I'd like to play through the two games (EE, no other mods, no exploits), and I'm wondering what people think is the best way to experience the most that both games have to offer. Any tips as to which class/race/alignment to pick, which NPCs to pick up along the way, etc.

Thanks in advance!

I'd say if you want the most out of the saga, play the kit you like, bbut you mention not playing solo but party.

If you want a party, then hire EVERYONE, do their quests, and pick late those you want to keep (I mean get ASAP 3 permanent NPC in BG 1,and 4 in BG 2, and rotate out the rest until you have no more). This way you get all the character-related quests.

Another way is, if you are all into "character interactions" to get you permanent troop ASAP, and enjoy all the random conversations. Though the game/s are so long you'll get bored of thatvery soon, but that's just me.

Alignment doesn't realy determine anything aside paying attention to reputation, and a couple of items/interactions.
If you want to play goody-good, then be Lawful Good, and get that sword in the Watcher's Keep. Nothing much comes out of it.
Evil allows a couple of extra options (working for Firekraag, working for demns in the Watcher's Keep teleport maze level).

Choosing your class is determined by a) what you want b) what your party will be c) what you want to try out.
For the later it mostly means gaining the Use Any Item HLA, so you can put any item in use and see what it does (or just get an NPC pick that).