While the question you ask is theoretically simple, there are a lot of factors that need to be taken into account when dealing with this. The difference between the two builds isn't just "Oh, I'm going to change out a couple of levels and have basically the same character!" There's a lot of differences, and there can be a lot of things that change between the characters. For simplicity, all of the characters will share the same race and ability scores. This means that I can calculate the same attribute bonuses how I like, and basically make the character have the skills we need.
The first decision is to assume we are going a strength-based character. We could do a bow Fighter, and it would work well especially with the long-range of the bow, but I am still trying to max out the Champion aspect of the Fighter/Wizard, and using a Str-Based character with a Greatsword means that the critrange of 19-20 is helped out that much more by 2d6 melee over 1d8 ranged damage from a Longbow.
The goal for the Knight is to get to basically max Str/Int for damage purposes, and then either 18 Con + Tough for max HP, or 20 Con for 25 less HP than the tough mode and gain +1 Con Mod.
For the Fighter/Wizard, we want to maximize damage. While this class should still have a fair bit of melee ability, we need to maximize the offensive abilities best we can. Trying to max out Str/Int first is good, and then having the rest of the ability score increases (which there are only 5 of) to round out the HP gives the Fighter/Wizard a significantly worse HP.
We are going to use a Mountain Dwarf. Since we only need Int, Str, and Con as our stats to max now, since Wis, Cha, and Dex don't matter, we want to maximize those values. Mountain Dwarf gives +2 to Str and +2 to Con, so we can start out with a 17/8/17/15/8/8 character. With the Eldritch Knight, it's impossibly easy to get 11 points of Attributes out of the 7 attribute increases.
Itemization will be the same. They will both have 2 +1 Longswords/Shortswords and +1 Full Plate (for defensive purposes). This will give them max AC and max damage for now.
The target will be an Ancient Red Dragon. It has 22 AC, which to my knowledge is the hardest thing to hit at the moment.
From there, it becomes a simple measure of planning and building the characters.
Level 20 Mountain Dwarf Fighter (Eldrith Knight)
- Max HP: 244 (10 + 19*6 + 20 * 4 + 20 * 2)
- Proficiency: +6
- Strength: 20 (+5)
- Dexterity: 8 (-1)
- Constitution: 18 (+4)
- Intelligence: 20 (+5)
- Wisdom: 8 (-1)
Charisma: 8 (-1)
- 4th: Dual Wielding
- 6th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
- 8th: Tough
- 12th: Str to 20 (20/8/17/15/8/8)
- 14th: Con to 18, Int to 16 (20/8/18/16/8/8)
- 16th: Int to 18 (20/8/18/18/8/8)
- 19th: Int to 20 (20/8/18/20/8/8)
Two Weapon Fighting (Fighting Style), Second Wind, Action Surge (Two Uses), Weapon Bond, Extra Attack (3), Indomitable (three uses), Weapon Bond (Eldritch Knight), Improved War Magic (Eldritch Knight), Eldritch Strike (Eldritch Knight), Arcane Charge (Eldritch Knight)
- Spell Slots: 4/3/3/1
Feats: Tough, Dual Wielding, Heavy Armor Master
Main Hand: +1 Longsword (+12 Attack, 1d8 + 12 Damage (16.5 Avg. Dmg.), 2d8 + 12 Crit (21 Avg. Dmg. Crit))
- Off Hand: +1 Longsword (+12 Attack, 1d8 + 6 Damage (10.5 Avg. Dmg.), 2d8 + 6 Crit (15 Avg. Dmg. Crit))
AC: 20 (+1 Full Plate, Dual Wielding)
Rich is probably going to be able to deal a fair bit of damage. Each single Main Hand Longsword strike will deal an average of 16.5 Damage on a hit. Against a Red Dragon, that's a 55% chance to hit (and 5% to crit), so 9.3 damage per swing accounting for accuracy (and crit). The offhand is 10.5 damage on average, so a 6 damage per swing accounting for accuracy (and crit). That means every attack action is going to be 15.3 damage per swing. Considering that Rich will get 4 attacks per turn, this is 61.2 DPR on just a basic set of attacks. With an Action Surge, that damage jumps to 122.4, which is approximately 22% of an Ancient Red Dragon's max HP. Two turns right after each other, Rich Alone could easily bloody the Dragon with slightly lucky rolls.
The main hand weapon used to be a +1 Greatsword. Compared to the current build, it did less damage on Rich's turn (10.8 damage per Attack action, 43.2 damage per turn vs. 15.3 per Attack Action, 61.2 per turn), but had a higher reaction -> opportunity attack damage. This is because it had an average damage of 19 Damage (Crit 26). These extra points of damage could stack up well, but to be honest, the lack of 2.5-5 points of damage on the 1 reaction per round loses out to the heavily increase DPR thanks to the dual wielding. The Greatsword also had a higher Improved War Magic damage, thanks to being able to attack only with your main-hand weapon. Again, while the loss of 2-5 damage per Spell -> Attack is known, the heavy increase in overall DPR from the multiattack functionality of Fighters is much more useful.
Fig Tier Wiz
Level 20 Mountain Dwarf Fighter (10)/Wizard (10)
- Max HP: 184 (10 + 9*6 + 10*4 + 20*4)
- Proficiency: +6
- Strength: 20 (+5)
- Dexterity: 10 (+1)
- Constitution: 18 (+4)
- Intelligence: 20 (+5)
- Wisdom: 8 (-1)
Charisma: 8 (-1)
- 4th: Heavy Armor Master (Str to 18 (18/8/17/15/8/8))
- 6th: Str to 20 (20/8/17/15/8/8)
- 8th: Int to 17 (20/8/17/17/8/8)
- 14th: Int to 19 (20/8/17/19/8/8)
- 18th: Con to 18, Int to 20 (20/8/18/20/8/8)
, Second Wind, Action Surge (One Use), Improved Critical, Extra Attack, Remarkable Athlete, Indomitable (one use), Great Weapon Fighting (Additional Fighting Style); Arcane Recovery, Arcane Tradition, X Savant, 3 Arcane Tradition Features (Arcane Ward, Projected War, Improved Abjuration || Sculpt Spell, Potent Cantrip, Empwered Evocation)
- Spell Slots: 4/3/3/3/2
- Feats: Heavy Armor Master
- Main Hand: +1 Greatsword (+12 Attack, 2d6 + 12 Damage (19 Avg. Dmg.), 4d6 + 12 Crit (26 Avg. Dmg. Crit))
- Feature: Defense (Fighting Style)
- AC: 20 (+1 Full Plate, Defense)
- Main Hand: +1 Shortsword (+12 Attack, 1d6 + 12 Damage (15.5 Avg. Dmg), 2d6 + 12 Crit (19 Avg. Dmg. Crit))
- Off Hand: +1 Shortsword (+12 Attack, 1d6 + 6 Damage (9.5 Avg. Dmg), 2d6 + 6 Crit (13 Avg. Dmg. Crit))
- Feature: Two Weapon Fighting (Fighting Style)
- AC: 19 (+1 Full Plate)
So here I've given out 2 builds for Fig that both can work. We only have 5 Ability Score increases, so it is much more difficult to max out Str and Int and get a high Con score. Technically, Heavy Armor Master can be exchanged for any feat that gives +1 Str, Con, or Int, but the DR 3 on non-magic weapons seems like a huge deal.
Option 1 is what I had originally come up with for Fig and Rich. The idea is to max out damage thanks to Str and Int, and use a Greatsword for combat. Because of that, you can take the Defense Fighting Style for +1 AC. The damage for this character is 10.8 Damage per attack accounting for accuracy against an Ancient red Dragon, and therefore a 43.2 Damage per turn.
Option 2 is more like Rich, wanting to use two weapons. The best/only options for max 1-handed light damage is Shortsword basically. As with Rich, the damage is greater over the course of a round than using the Greatsword because of the multiple hits you get in per attack, but you have a lower amount of damage on a reaction, by about 2 points (3.5 points on a hit, 7 points on a crit). This build sacrifices 1 AC to get an increase of damage at close range.
(As an aside, Option 2 could also possible use an 18/8/18/20/8/8 final ability score with 2 +1 Longswords and Dual Wielding. I checked the numbers, and while you do get +1 AC thanks to dual weilding, you actually lose damage overall. This is because while you do get to go from 1d6 weapons to 1d8, you lose +1 to your Attack Modifier and +1 to your damage. The mainhand would be +11 to attack, 1d8 + 11 Damage (15.5 Avg. Dmg), 2d8 + 11 Crit (20 Avg. Dmg. Crit)). As you can see, the main hand would basically lose 5% on normal attacks, while the Average Crit damage increases by 1. This is the same for the off hand. The only difference is basically that you lose 5% to-hit to gain a marginal benefit when you crit.)
So right off the bat we see that Fig is lacking in a couple of areas compared to Rich. He's already at a much lower HP thanks to the levels in Wizard only giving +4 to HP per level instead of +6. In addition, the lack of ability score increases means that hitting 18 Con + Tough or 20 Con is almost impossible without sacrificing damage. With 60 HP less than Rich, it's a fair bit of damage that is not turned lethal.
Second, the lack of Action Surges and Extra Attacks means that Fig won't be able to attack more than twice a turn. This is huge when you compare the two characters. The lack of Dual Weilding for Fig means no "natural" dual wield longsword, and therefore a fair bit of loss of damage. On top of that, you can only deal ~28.2 damage per turn, ~56.4 with your single Action Surge. The damage that Fig is going to be putting out is on the level of a 10th level fighter, not a 20th Level Fighter/Wizard.
Speaking of the Wizard side, you are still not a super-powerful spellcaster when it comes to spells. What you give up from the Wizard Side to "splash" 10 levels in Figher means that you only have access to 5th level spells. While not an issue for the most part, 5th levels spells will need to greatly make up for the damage deficit that the lack of extra attacks the fighter should have incurs. You also "only" have 2 5th level slots and 2 4th level slots over the Knight. I put "only" in quotes because while I can attempt to downplay those slots, they represent a level of power the Knight cannot achieve. If we are talking about pure damage, the Knight can cast 1 4th level spell, but will probably want to prefer to use his spells as support spells to further lock-down his targets while he beats the heck out of them. The Wizard can use those 4 extra slots to provide a lot of utility to the party.
For comparison of the spell casting abilities, we need to look at the damage between the classes. Because spell damage is only based onthe spell, not your For now, I think the best option to check for spell damage is going to be the Spell Damage Comparison Chart put out by Rhaegar83 on /r/dndnext.
Looking at the spells, it's clear that no 5th level spell can match the single-turn damage of a 20th Level Eldritch Knight attacking the crap out of a single target. While you do get access to some nice 5th level AoE spells like Cone of Cold and Flame Strike, the Eldritch Knight can easily hit 2 targets for similar Damage if he is able to get between the two using his move.
The end result is that Rich, the Eldritch Knight is going to end up tankier (HP and AC) and able to deal more single-target damage thanks to his quad-attack. Fig is going to be able to deal more AoE damage, single by virtue of having access to 5th level spells, as well as 2 more 4th level spells slots. Fig is also going to be able to be more of a controlling presence on the battlefield, if he is going to take at least some support/control-y spells.
“There is no point in doing this [from a game mechanic perspective].” The sorcerer is just the stronger class, and even the feature-heavy first level of bard does not compare to simply having better spells sooner (see 1. Spellcasters should not multiclass in this answer for more details). Thus, the best mix of nine levels for bard or sorcerer is Sorcerer 9. That said...
The best way to multiclass bard and “sorcerer” is to not have any sorcerer levels at all, but rather take the sublime chord prestige class from Complete Arcane. This class requires Bardic Mustic and 3rd-level spells, but rather than progress bard spellcasting, it has its own spellcasting with 3rd- to 9th-level spells from the bard and sorcerer/wizard lists. This spellcasting is Charisma-based and spontaneous. It also progresses bardic music, and gives several special “magic themed” songs.
So a Bard 10/Sublime Chord 10 casts as a 10th-level bard and also has separate spellcasting with spells of up to 9th-level that come from the sorcerer/wizard spell list (or bard list). It has the ability with music of a 20th-level bard, except some of the songs are changed to be more “magic themed.”
Since spellcasting is the only sorcerer class feature aside from the familiar, having spontaneous Charisma-based spellcasting off of the sorcerer/wizard list, and then taking the Obtain Familiar feat, makes you effectively identical to a sorcerer. But this progression is much smoother, you end up with level-appropriate power at higher levels, and sublime chord is really cool. The only problem here is that, before 11th level when you take your first level of sublime chord, you have no mechanical representation of being a sorcerer. But bard and sorcerer spellcasting are fairly similar, and you can take Obtain Familiar at Bard 1, so it should be easy to continue to call yourself a sorcerer at lower levels.
Worth mentioning: Champions of Valor has a variant paladin, the harmonizing knight, that gets Inspire Courage +1, 1/day instead of at-will detect evil at 1st level. In the Forgotten Realms, this requires you to worship Milil, a goddess of music; in other settings, it would have to be adapted to some appropriate patron. Anyway, all paladins get the excellent Divine Grace at 2nd level, adding Charisma bonus to all saving throws. As such, Paladin 2/Bard 8/Sublime Chord 10 becomes an excellent variant on the above build: you trade 1 daily use of Inspire Courage for full martial weapon proficiency, a bit more HP, +1 BAB, and adding your Charisma bonus to all saving throws. Since your Charisma should be high, that is a very nice bonus. Adding paladin is not an option for all characters, of course, but if it is, do consider it.
For spellswording as a bard, whether you dip paladin or not, I strongly recommend the Snowflake Wardance feat from Frostburn, if you have that book. Other excellent options include the harmonizing weapon property and crystal echoblade weapon from Magic Item Compendium.
Finally, if you have Tome of Battle, taking a level of crusader for the Song of the White Raven feat is an awesome option, dramatically amping up your physical prowess while allowing you to start performing Inspire Courage as a swift action. It also opens up the interesting possibility of using the jade phoenix mage prestige class to advance sublime chord spellcasting, which would be ideal. Note that Paladin 2/Bard 7/Crusader 1 still just qualifies for sublime chord, too, if you want to do both. I recommend taking the crusader level at precisely 9th level, so you can simultaneously take Song of the White Raven, and have Initiator Level 5 so 3rd-level maneuvers and stances are available to you.
Generally speaking, race isn’t all that important; human is probably your best bet just because bonus feats are awesome. But anything without a penalty to Charisma or Constitution is probably fine (bonuses to Charisma are really rare and always paired with a penalty to Constitution, so there is little to be gained there). Even penalties to Charisma or Constitution are bearable, but why would you?
However, in the case of anyone with bardic music, the benefits of being a dragonblooded race have to be mentioned. And since there’s a dragonblooded human race, silverbrow humans from Dragon Magic are almost-certainly your best option. They trade the humans’ bonus skill point for the Dragonblood subtype, which among other things qualifies you for the excellent Dragonfire Inspiration feat from the same book. Highly recommended.
I'd like to begin by noting that multiclassing is a dangerous choice, especially for a new player. If you choose any single character class, the 5e rules will generally guide you to a reasonable character that is viable in combat. If you multiclass, it's possible that some of the levels you take will be "deadweight" levels which won't help you significantly in combat, and your function in combat will be that of a lower-level character.
In particular: you've told us that you'd like to play as a sorcerer. As a sorcerer, your function in combat will be to cast spells, and your fighter level will not make you any better at that. So, although your survivability as a multiclass character will be higher, your damage output will be that of a second-level character in a third-level party, and that's a significant drawback.
So the general advice we have for players of D&D 5e, especially for inexperienced players, is to not multiclass.
Let's now talk about details.
I don't think multiclassing serves you well here.
Now: third level is an unusually bad level to give up to multiclass to fighter, because the sorcerer gains a lot from their third level -- they double their spell slots and unlock a new level of spells. But fourth level is unusually bad too, because the sorcerer gains a lot from that one -- they get +2 to Charisma, which grants +1 to-hit and to save DC. Fifth level is especially terrible because the sorcerer gets fireball then, which can alter the course of whole combats.
What I'm saying is: giving up a level of sorcerer will hurt you a lot.
I'd like to idly note: if you did want to improve your AC, you might also consider multiclassing to cleric. Most of the cleric domains will grant you proficiency with heavy armor -- you wouldn't get that multiclassing to fighter. Cleric will also grant you spell slot progression, so a sorcerer2/cleric1 would have the full six spell slots including two second-level spell slots. It's almost worth it, except that cleric doesn't grant you progression in sorcerer spells known, so you'd still be a level behind in most of the ways that are important.