Paladin 2 / Wizard (abjurer) X
You said you are willing to take at most a 2 level dip in another class for proficiencies. Have you considered taking these levels in Paladin? Assuming you want to keep the "I'm a Wizard!" flavour, taking your first 2 levels as a Paladin of the god of magic (assuming there is one in your campaign, of course) isn't really out of character. So here's my take on this build. I'll mostly cover melee options, since this is what the question is about, but you should still be able to do anything that another wizard could do, just 2 levels later. This gap can be great at certain levels (Fireball is a great damage upgrade at level 5), but it shouldn't be noticed too much on single target damage (Smite!). It's not the strongest build out there, but it's definitely good enough to be enjoyable!
Benefits of 2 paladin levels :
- Proficiency with all weapons : opens up many solid weapon choices
- Proficiency with all armors and shields : although dex is definitely the best stat in 5E, a melee wizard will require spending spell slots on defence to keep up. Being able to wear a full plate and a shield gives great defensive benefits and costs no daily ressources.
- 6 more HPs : While 2 more level wizard levels would've given you 4 more arcane ward HPs, those only replenish on a long rest or by spending spell slots. Arcane ward is an awesome feature, but real HPs trump it anytime.
- Lay on Hands : Yay for 10 free HPs per day! More than makes up for those 4 lost ward HPs, and can be used on other people, too.
- Fighting style : There are many good options here, but I'd go for protection or defence to increase your utility/survival. Great weapon fighting is a good offensive option if you forgo your shield and grab a great sword, especially since you're going to stack lots of dice on those attacks. This can be somewhat replaced by the Elemental Adept feat, however.
- Access to 1st level paladin spells : One thing you'll have as a wizard is lots of low-level spell slots rather early. Smite spells might not always be the best options, but they can add good damage and other effects on your melee attacks. Paladin also has a few abjuration spells, which gives you back some arcane ward HPs, but the good ones are higher level, sadly. If you have the warcaster feat and can handle your constitution saves, Shield of Faith is also very good as a low-cost defensive boost. While Cure Wounds is a paladin spell, you can cast it using your full spell slots, which means that, although not a cleric, you still have access to some great healing powers if they are needed.
- Divine Smite : Convert spell slots into damage. Given that you can elect to smite after you know the attack hits, and that there is no save for this damage, it's a great way to increase your melee damage. Especially good if you crit...
That being said, given equal gear, you should have the same AC as any other tank, since you're wearing the same stuff they are. As for your damage, the new Sword Coast Adventure Guide cantrips are there for you. Sure, they're not going to out-damage a fighter or a rogue, but they're still very good damage options. Since they require a melee weapon attack, they also trigger class features such as Divine Smite, so you don't miss out too much. Having two attacks could technically mean 2 smites per round, but the added damage of your cantrips will compensate for this quite decently.
- Green-Flame Blade : when fighting multiple enemies, this attack basically gives you the benefit of a second attack without costing anything. Since you're not going to have multiple attacks from class features, this is pretty much as good as you're going to get.
- Booming Blade : Assuming you can make sure your opponent moves willingly each round, this cantrip will give you the same benefits as GFB except on a single target.
Other thing to note is that the secondary damage from these spells automatically hits, no save and no attack roll required, which means that you can strike a minion to land free damage on the BBEG. These spells will thus give you decent damage round after round, leaving you with all your other spell slots to use as you see fit. Your 2 levels of paladin will give you the opportunity to convert your low-level spell slots into more damage as an option, and your wizard powers will allow you to turn these same spell slots into defensive abilities. Shield will grant you +5 AC for one round, which is awesome if you attract the BBEG's attention (especially given that your AC is already high). In Elemental Evil, there is also the Absorb Element spell, which gives you resistance against a single element, also for one round. This gives you defensive options for both physical and most magical attacks. As an interesting addition, both of these spells are abjuration-based, and will thus heal your ward for 2 HPs every time you cast them.
And after all this, you're still an almost full wizard. Your access to spells and features will be delayed for two levels because of multi-classing, which shouldn't reduce your melee effectiveness, but you will still have access to 9th level spells and have spell slots as a 19th level caster. While the Signature Spell capstone is good, the real treat is at wizard level 18 (character level 20 for you), so you don't miss out on much. Indeed, spell mastery will allow you to have 2 low-level at-will spells, which gives you infinite Shield spells, for instance. This should allow you to use your slots for more offensive options, since your defence is free. Infinite Shield spells also means that your arcane ward is always topped off after every combat if you take a minute or so to recharge it after each fight.
Your melee power also scale quite decently if we compare them to a tank fighter (longsword + shield, defensive fighting style) of equivalent level :
- Level 1-4 : Fighter [1d8 + str] vs GFB [1d8 + str + (int secondary)]
- Level 5-10 : Fighter 2x[1d8 + str] vs GFB [1d8 + 1d8 + str + (1d8 + int secondary)]
- Level 11-16 : Fighter 3x[1d8 + str] vs GFB [1d8 + 2d8 + str + (2d8 + int secondary)]
- Level 17-19 : Fighter 3x[1d8 + str] vs GFB [1d8 + 3d8 + str + (3d8 + int secondary)]
- Level 20 : Fighter 4x[1d8 + str] vs GFB [1d8 + 3d8 + str + (3d8 + int secondary)]
Of course, a fighter has abilities to increase his damage, but so do you (thank you, Divine Smite)! Since the fighter has more attacks, he's more likely to deal some damage each round, whereas your damage all hinges on your single attack landing. This will be especially important if the fighter can get his hands on a weapon that deals extra damage (like a flame tongue), an ability that would be mostly wasted on your single attack per round. The fighter also only requires only one good stat to be 100% effective, while you will need both strength and intelligence. Since your intention is to go melee, I'd consider maxing strength first, especially considered you'll use your spells mostly defensively or in a utilitarian fashion (so lower save DCs are not as bad).
Feats to consider :
- Warcaster : For pretty obvious reasons (casting and adv. on constitution saves). Combined with Booming blade, it's also a boost to damage: when an enemy tries to get away, you booming blade it, which will deal almost double damage if it hits, as the enemy will likely leave the area right after. If they don't, well, your party gets to kill them anyways.
- Shield Master : If you are using a shield, it's a solid defensive feat.
- Heavy Armor Master : +1 strength, reduce most physical oncoming damage by 3. The drawback is that it doesn't work on your arcane ward, as it has it's own HP pool. As such, any damage the ward takes is not "damage that you take" and does not trigger this feat. Still a great option for a tank, especially one that has a wizard's HP pool and requires strength for his attacks.
- Elemental Adept : You're going to cast a lot of fire cantrips, and this feat allows you to ignore resistance and to treat 1s as 2s. This feat does not specify that only the fire damage dice are affected, so it works on your weapon die, your GFB dice, but also on your added Divine Smite dice. Definitely not bad.
- Mage Slayer : Disrupt enemy spellcasters from melee range. The attack granted by this feat is not as strong as a full opportunity attack (as warcaster will allow for a GFB or BB cantrip), but if you can prevent them from casting, it has a lot more utility. Advantage on saves isn't exactly bad either.
Other things to consider :
- Transmuter instead of abjurer : possible proficiency with constitution saves or resistance to one element is a good trade-off for a melee wizard. Not sure that I'd sacrifice arcane ward for it, though. Since Arcane Ward has it's own HP pool, all the damage it soaks isn't counted towards the concentration save DC. Advantage from warcaster should be more than enough, especially if you have some stat bonuses.
- Bladesinger instead of abjurer : A tough one... Since you're going to use cantrips for offense, you're wasting the extra attack feature with that build. But... a dex-based character wearing light armor will have only 1 less AC than a plate wearing character (equal if you cast mage armor instead), so your bladesong will leave you with a net +2 to AC during a fight once you get 20 intelligence (since you're also forgoing your shield). Even if not going for a strength build, you lose none of the advantages of a paladin (smite, armor proficiencies, healing, spells, etc.), but also gain Intelligence bonus to concentration saves and increased combat speed, as well as overall better skills since dexterity is the best stat. I'd personally favour bladesinger, but both options are very interesting.
Part of the benefit of the arcane discovery alchemical affinity says, "Additionally, you may copy spells from an alchemist's formula book into your spellbook just as you could with another wizard's spellbook." So Magical Marketplace's author, at least, believed it normally impossible for a wizard to copy into his spellbook a formula from an alchemist's formula book.
(Further, the alchemist's supernatural ability alchemy says, "A wizard, however, cannot learn spells from a formula book," and while Pathfinder isn't as clear as it should be when tossing around a very pregnant word like learn, Adding Spells to a Wizard's Spellbook nonetheless says, "If the [Spellcraft skill] check fails, the wizard cannot understand or copy the spell" (emphasis mine). The text continues immediately after, saying, "He cannot attempt to learn or copy that spell again until one week has passed" (emphasis mine). Parallelism makes understand and learn synonyms here, so a wizard, even after waiting a week, still won't be able to make that Spellcraft check to copy a formula because a wizard can't learn spells from a formula book in the first place. Note that I don't like this ruling as much as I like the anecdotal evidence from Marketplace because whenever the GM must make a ruling based on a rule's syntax there's usually more than one possible reading, making disagreements and hard feelings a possibility, so make of this what you will.)
A house rule could allow copying formulas into spellbooks anyway, despite the existence of alchemical affinity
A GM that makes a house rule that allows the PC to add his handful of appropriate alchemist's formulas to his wizard's spellbook (and still requires time and money to transcribe them) needn't worry that the PC is attempting some kind of crazy power-grab. Under such a house rule, a wizard who, late in his career, decides to take levels in alchemist trades versatility for power… and makes a completely uneven trade.
Advantages of a 3/3 multi-class:
More known spells than sorcerer (6). [Same number of spells as a wizard (6) with the same starting INT]
Increased spell selection over sorcerer (6)
More sorcery points than wizard (6).
Ritual spells (not available as a sorcerer without a feat)
Metamagic can be applied to spells being cast via familiar
Disadvantages of a 3/3 multi-class:
Multi Ability Dependant (MAD, requires Int and Cha, on top of Con and Dex).
Does not know 3rd level spells (7th level characters know 4th level spells)
Know less spells than a Wizard (6)
Fewer feats/ASI (although a 4/4 has the same ASI as a single class 8)
Fewer sorcery points than sorcerer (6)
Delayed feature progression in both classes.
In my opinion (as a player with 2 years exp and a DM with 1 year exp), the MADness and lack of 3rd (and at the next level, 4th) level spells makes this a sub-optimal character.
Additionally, you are short a ASI/feat for a level, and will be less effective with your sorcery points because you only have 3.
It won't be so weak that you can't play it, but you will feel less powerful than the other dedicated casters (and maybe the half casters as well) in your group. if you are trying to build a strong character, you are better off going full in either class. Sorcerer will give you more power, but less options, while wizard will give you more options, but slightly less power.
You might consider only taking a 1-2 level dip into wizard, and mainlining sorcerer instead. Then taking ritual spells, and spells like mage armor and shield that don't require INT to cast. This will get you the increased flexibility of a wizard, but eliminate some of the MADness and let you get that ASI right away (you still need 13 INT to multi-class). It will also only delay your spell progression by 2 levels, instead of 3. You will have less pure power than a pure sorcerer, but significantly more flexibility. However, if the ritual spells are the only thing you are looking for, then taking ritual caster will be much more efficient than 1-2 levels of wizard.
In general, multi-classing full casters is a bad idea (even those who share a spellcasting ability). Every level you don't take in your main class is a level longer before you get spells for a slot that you already have. On a pure damage comparison, casting a 1st level spell with a 3rd level slot will do less damage than casting a 3rd level spell with that same slot. And that speaks nothing to the utility of the higher level spells. One or two level dips into another class are occasionally worth it, depending on how front loaded the class is, but any more than that and you start significantly delaying your spell progression.
Yes, you get more cantrips for multi-classing, but you're only going to use one or two most of the time, and by 6th level, you have 10 spell slots. You will probably only have 15 rounds of combat in a day, so you won't need your cantrips that much.
To directly answer your question (is there a new ability that the multi-class gets that neither original class gets):
There is no novel feature (like your examples of action surged spells or spellcasting in armor, although you could take feats to be able to do that as a pure wizard) that you get from multi-classing these two.
However, as per this question, metamagic can be used on non-socerer spells, so you can get new combinations by modifying your wizard spells. For example, if you took Protection from Good and Evil as a wizard spell (not available as a sorcerer), you could extend it using metamagic to have a 30 ft range. While this is technically a new feature, it is very situational, and probably not worth the multi-class.