One thing to note is that the new DSi doesn't have a "slot2" opening for GBA cartridges. Therefore, emulating GBA games might be harder (haven't looked into if this is possible yet). You need a special "NDSi" TopToy or Acekard chip to work with it. (Their is an R4 clone for the DSi as well).
I have 3 TTDS carts for my standard DS lites (for me and my 2 sons), and all have worked great. I just ordered another today as a spare - less than $6 shipped - you can't go wrong with that. Everyone seems to day "buy the Acecard2, its way better", but my Acecard2 went bad after 1 week, while my cheapo TopToy carts (and 2 N5 carts I also bought) have been working great for a long time.
Emulators:
SNES: You can play some of the SNES roms using SNEmulDS, but the more advanced SNES games, such as MarioRPG or Chronotrigger, don't work.
GBA: The best way to play GBA games on the DS Lite (or original DS) is to get a slot 2 cart. I have an EZ-Flash 3 and a Supercard. The EZ-Flash III works great (best compatbility) but is a bit of a pain to work with. The Supercard is easy to work with, and has a much larger capacity, but some games are hard to get working on it.
Sega Genesis: Never tried this, but they exist. From what I know, not all games work
NES: This system can be emulated quite well on the NDS. The best program is NesDS, which adds new features to the old NES games, such as real-time save, and "rewind"
Other: You can also emulate original Gameboy and Gameboy color games, Gamegear games, and several other older platforms.
The TopToy takes an SD card - I have an 8GB one in mine. The games take up about 2 GB, while the rest of the space is filled with music, and a few movies. If you want to use the DS Lite for movies and music (along with gaming) then I would recommend a 8GB card. If you use it just for games, then 4 GB is fine.