It will take some work, but it's perfectly doable.
Enemies with special defenses
The players won't have access to see invisible, flight, magic weapons, or ghost touch. Therefore, if you include enemies that can fly, or have DR/magic, or are intangible, these will be huge challenges, and may be downright impossible. The easy solution is simply to not include such enemies, but the more interesting way is to treat them as nigh-invulnerable enemies that the heroes will have to figure out how to defeat. In a normal Pathfinder game, a ghost is a normal critter with a +2 CR template pasted on top, and requires some minimal preparation to defeat. In a low-magic Pathfinder game, a ghost is a mystery: why did they become a ghost? How can you persuade them to pass on, or at least let the party pass by peacefully? Can you persuade the local priest to perform an exorcism, and will it even work? Instead of "find monster, insert fireball," these types of encounters are now role-playing challenges, because they can't be solved any other way!
Alternatively, you can simply strip out the special defenses from enemies. Pathfinder assumes you have level-appropriate counters to special abilities anyway, so by removing those special defenses, you aren't going too far from the original intent. Adjusting the CR is left as an exercise for the GM, because it's going to take a fair amount of trial-and-error to determine what the right balance is.
Fixing armor class, and other numbers issues
Pathfinder assumes that both attack bonuses and defenses will be augmented by magic items. This partially balances out if the players don't have magic items, but consider giving everyone a +1 bonus to AC and all defenses every four levels. Don't make them pay a feat for it, just give it to them.
While you're at it, give your players bonus XP for the monsters they defeat, by calculating the XP as if the monsters were a higher CR. Since they're operating without magic, every encounter is going to be harder than what the DMG "expects" when it calculates XP per CR.
What will they do with their money?
Your players won't be able to buy gear that personally enhances their ability to make things dead faster, or grant them new solutions. If you keep to the normal loot rules, then the party will have far, far more money than they know what to do with. You have two options here: give them less money, or give them something to do with that money.
Let them invest in mercenary companies or land holdings. Let them become influential in the church, or their hometown, or even their country as their economic might and donations in the right places give them power that they would never be able to take with a sword. Favors in high places give characters some very powerful options.
Recovery after combat
This will require explicit house rules; you'll need to accelerate natural healing (heal a percentage of HP per day instead of a flat amount?), allow Heal checks to do much more than they normally do, grant the local clergy some extremely localized healing powers (they can heal people brought to their church, but not outside of their place of worship), and/or make this a political game rather than a hack-and-slash game.
If everyone's having fun, then it's a good game. It doesn't matter if the characters aren't optimized: as long as they feel like they're making a difference in the world and they're enjoying the game, then you're doing it right. The characters will be balanced, more or less: they all don't have access to magic, so intra-party balance isn't as much of a problem. You'll see that the players lack all of the magic-based solutions that you'd expect in a normal Pathfinder game, and you'll select the enemies more carefully, but things will work out fine. Let your players know that things will be a bit different than normal, and your players will go along with it; they requested this kind of game, after all.
“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
Best Answer
Your goal is to increase the will save.
My recommendation is to make the race Elf or Half-Elf or Dwarf. Elves and Half-Elves recieve +2 bonus on enchantment spells which covers most of the Mind-affecting spells. Dwarves have +2 bonus against all spells and a racial +2 wisdom to increase their will save.
As a class, I would go with Druid or Cleric if you can because those classes have a good will save progress and their main Stat is Wisdom which will net a very good Will save bonus.
Top those with Iron Will and Improved Iron Will feats and that character will be very strong against mind affecting spells unless the caster is ridicoulusly high level or your rolls are very unlucky.
As a result, assuming you have a level 2 half-elf Cleric, with 16 Wisdom you have +10 Will(+3 Cleric,+2 Half-Elf,+2 Iron Will,+3 Wisdom) save versus Enchantment spells with a chance to reroll one once per day without any magical aid.
There are a lot more ways to incrase your mental resolve against mind affecting spells. Traits, items or playing a Paladin like Rob mentioned in his answer but It's difficult to deduce what is allowed in your campaign or what kind of character is suitable. I intentionally left out domains and Paladins because they are restricted to alignment or the Deity. This is quite a general approach that is suitable to any character.