“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...

Sublime Chord

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.

Spellswording

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.

Race

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.

See Also

• Reasons to multiclass in 3.5 (already linked, but worth checking out in general)
• Are 3.5 Bards mechanically weak?
• How to optimize a Gishy [Fighter/Mage] Character?

Flavour over Fluff

The key to understanding 3.5e/PF classes is that while they have evocative names (Rogue, Fighter, Assassin, Archmage) and a big pile of writing at the front that says it tells you what they do, often the mechanical effects of the class are not in line with the fluff - and another class might make a better version of the concept you are going for than one with a name closer to the concept.

For example, if a player came to me saying they wanted a character with innate magic they didn't know how to use gifted by dragon blood, and were struggling to control, i'd say Warlock without a second's thought. The Sorcerer class is word-for-word the fluff of what they are describing - but the mechanical abilities are extremely difficult to theme as 'innate, not-controlled magic' - it takes a skilled roleplayer to do so, and one who knows their way around the spell list. Warlock, though, is blasting people with raw magical energy and a few tricks that themselves are weird and not entirely what you might want. It's almost the ideal 'guy with magic who has no idea what he is doing' class, if you ignore the fluff entirely.

Fluff is the writing on a class that has no mechanical effect. As only the mechanics of a class actually translate into the game - your character's story and explanation for his powers supercedes what is written on the class - if you are concerned with flavour over mechanics, oddly, the first thing you have to do is look at the mechanics of any class you are thinking of applying to your character. As those are the things you can't change.

While many people are fine to use the existing fluff of a class, people heavily concerned with flavour will often want to change things up, refluff, or rewrite it to suit their specific character. As the only things you can't change are mechanics, those are the parts to pay attention to - because they are the ones that can stab your flavour in the back.

How To Make An Assassin In PF

Ninja. Ninja has a very eastern feel to it's writeup, but if you look at the abilities it grants, you quickly see there's nothing eastern about the effects. A shadow warrior concerned with stealth and striking swiftly from darkness, the ninja is much more like an assassin than the rogue - but a specific kind of assassin, not one who strikes from every direction and works the social aspect, but one who relies on physical and magical ability to climb up the side of an impassable castle and kill a king in his bed before disappearing into shadows and smoke just as the palace guard burst in.

Rogue. Rogues have the option to pick up some ninja abilities, as ninjas do rogues, but the cross-pollination is not as heavy as you might think, as the iconic (and defining) mechanical effects of the class remain class-specific. Rogues have the option to be sneakier, and dirtier than ninjas, which gives them more power due to the ability to have an indirect, or 'Face' like approach, but gives them slightly less raw 'teleporting through shadows murdering people'.

Vivisectionist Alchemist. This is the most assassin-like base class. You kill people, have a deep and enduring knowledge of anatomy, and use poisons. You also make magical potions that you use to empower yourself to kill people. It's the closest to famous fantasy-assassins, from the Witcher to Durzo Blint to FitzChivalry or Chade. Knowledge of secret herbs in fantasy = alchemy, and it and poison are hand in hand.

Mixing or matching these classes with a splash in some others could work, notably Fighter, Swashbuckler, Barbarian, and to a lesser extent, Inquisitor/Magus, both to represent a higher level of martial ability, or add more 'tricks' up your sleeve. Note that Magus/Inquisitor add martial ability, as does fighter - swash and mostly barbarian add tricks.

Overall, those three base classes are what you are going to be spending levels in to be a fantasy assassin. Vivisectionist gives you the most bang for your buck, but the other two are also fine if you know what you're getting into and dance past the trap options built into both classes.

Theoretically, a monk could be an assassin, with a decent level of monk mixed with rogue, and relying on Stunning Fist and feats that change/boost that ability, but that's a very specific flavour.

Prestige Classes.

Assassin. Assassin's a bit of a poser, because it does it's very best to straitjacket you into acting a certain way as hard as it can. But mechanically, it does only a very few things for you. Sneak attack progression, death attack, true death, and hide in plain sight. The first is self-explanatory, and most 'sneaky' prestige classes are immediately noped out of because they don't grant it. Death Attack is.. interesting. We'll get back to it. True Death is completely boring - so many low level spells, even alchemist's fire, do similar things. There are mechanisms for this not tied to 4 levels in a prestige class. Hide In Plain Sight is great, and fits your fluff perfectly - sadly, it requires level 8, and Assassin requires level 5 to get into - so you're getting this at level 13, aka, well into the game if not by the time the game is over.

Death Attack is really what we come back to. There are optimization resources for magic items, feats, and abilities that increase the DC - most of them are 3.5e. The existing DC will not be very high. Even if you pimp out your Int, and enter Assassin as soon as possible, no monster will die to it except caster-types - and no fighter-type npc will die to it, unless they are not wearing rings of resistance or the like at all. It's a huge let-down when you try this, and it doesn't work, so if you want it, you'll need to really invest in it, not just your character's resources, but also your time in researching how to improve it. It's a weak ability, so it's no small potatoes to try to improve it. On the gripping hand - poisoned dagger full attack with sneak attack from the shadows will likely kill anything that would die to that DC in the first place. So that's a question you have to ask yourself. What is this actually giving me?

Shadowdancer. This class is important because, while it requires 3 crappy feats to enter, it grants Hide In Plain Sight at first level. If it gave sneak attack progression (it doesn't), it would be a shoe-in. The abilities are good, and suit an assassin, especially once you strip the 'agent of the darknesses!' fluff from them. But it doesn't add to your murder ability, and that will start to hurt you as you level up. And the good abilities don't come thick and fast enough to make up for that. A 1 level dip costs you 3 feats and 1 level - but it lets you hide anywhere, and that is invaluable. This class + Assassin at the same time would be the ideal assassin prestige class.

I looked at all of the other prestige classes that seemed 'sneaky' but they were, to put it mildly, crap. Boring mechanics, and boring fluff. I really don't see any that even look slightly good.

I see three major options for you, to make an assassin in pf (as opposed to just a sneaky person) -

Vivisectionist Alchemist 5/Shadowdancer 1/Vivisectionist Alchemist 14

This gives you your magical poisoner/assassin, ala Durzo Blint or Chade. You can dip part of your later Alchemist progression into Assassin to pick up Death Attack, but it's that question again. Alchemist makes you progressively stronger as you get up in levels, enough to keep pace with casters. Rogue 3 gives you Improved Uncanny Dodge and Improved Evasion when combined with Shadowdancer 2, but probably isn't worth the dip.

Rogue 5/Shadowdancer 1/Rogue 14

This gives your silver-tongued, dark-eyed social assassin, who enters the enemy's bedchamber disguised as a maid. The lies and intrigue make you somewhat like the old 'Monster' field agent concept, with a soul as black as tarnished silver. You don't need to dip Assassin with this build, as several Rogue advanced talents are similar to Death Attack, the 'knockout blow' and 'deadly cocktail' being the most obvious. You can actually strip shadowdancer out of this if you want to spend the feats elsewhere - you rely less heavily on stealth than the other two builds, and can pick up Hide In Plain Sight from an advanced talent.

Ninja 5/Shadowdancer 1/Ninja 14

These builds all look pretty similar. And that's because in PF, very few prestige classes are worth it. Ninja has an advanced 'ninja trick' called 'Assassination' that uses word for word the same text as Death Attack. They also get many things that help get into position to assassinate, that the assassin doesn't get.

This is actually the closest to a knife in the darkness, a mundane assassin, that you'll find.