Any class, but only one, which is your favorite. This class can be any class or class combination, either multi, or dual, but it has to be legit and accessible only in standalone game without editors/hacks/cheats. I myself liked a lot an elf druid, and it has become my favorite character. Druids rock here.

Halfling Fighter/Thief. Backstabbing is fun.
Unlike in the Baldur's Gate games Druids are fun too.

My favorite has always been Dwarven Fighter/Cleric.

My favorite has to be any multi class where one of the classes is a fighter. Single class clerics, druid, thieves and mages (with the exception of specialist mages) are all pointless in IWD1.

is the multi-class option part of one of the mods, or something? Because I'm not running any mods, and I did not get the choice of a multi-class; it was either single, or strict dual.

Except human, any races can be multi-class

well, I feel like an idiot. 'Cause I definitely had that exact same screen when I created my party; I have an elf mage that coul have done with some fighting skills.

I just finished my first complete solo playthrough of IWD1 with a half elf fighter/mage/thief. While you won't need the thief component when playing with a party, an elf fighter/mage can really be a versatile addition to your party. As long as you keep stoneskin and mirror image active, he won't be needing much healing (salamander auras were annoying though), even when tanking on the front lines. Before, I neglected fighter/mages because I didn't like the idea of a front line tank wearing robes but he turned out much better than I thought.

I strongly suggest ditching the generalist mage for a fighter/mage, you won't be disappointed.

If you're lucky enough to get the Elven Chain or Kaylessa's Chain, your FTR/Mage can wear armor and cast spells. The pure mage will level faster, giving you access to higher spell levels sooner. Of course, this is moot if you're not finding higher level spells to memorize.

That's true, but if you're going pure mage, I would use a specialist. As long as your specialist can cast evocation and conjuring school spells, it's good. Opposition school spells can be worked around.

Make sure the race is not human if you want a multi-class character.
If this doesn't work, try to uninstall and re-install. Maybe some files were broken?