Haha, I got the wife a DS Phat for Valentine's Day last year. ;)
Mario Kart DS is good if you're into racing games. If you get a second DS (recommended!) you can do single cart download play (e.g.- one copy of the game, multiple players)... although tracks and vehicles are limited. It's still fun, though.
Bomberman DS (the first Bomberman DS game, haven't played the "touch" version) is a lot of fun with multiplayer, too. With a single copy of the game you and up to 8 other people can play head to head. It uses the microphone and touch screen pretty heavily for things like detonating your bombs. Some of the wildest games we played involved using an option where you had to talk into the mic to make your bomb explode... then all of us yelling at random while playing causing each other's bombs to explode at the most in-opportune moments.
Animal Crossing Wild World is fun. I tend to burn myself out on it after periods of playing obsessively, though. For people who are "collectors", it's a very addicting game. The ability to go online and swap stuff with other players is a nice addition, too.
We've got Brain Age too. I liked it but I never got into a habit of playing the game regularly.
I've heard good things about Castlevania: Portrait of Ruin and Final Fantasy III.
For playing older Gameboy games, the DS will do GBA games... and that's about it. Gameboy Color and original Gameboy games aren't compatible... if you want that you'd need to buy an actual GBA: SP.
If you get a GBA ROM cart (I bought one of these a couple years back: http://www.efa.cc/
), you can emulate original Gameboy games. I dinked around with it... but the nostalgia factor wasn't enough for me to overlook how dated the games look now.
Using a GBA ROM cart, you can also emulate an NES. I've had a LOT of fun with this; I played a lot of old school NES growing up, so it's like second childhood sometimes.
There are lots of places around and about that you can, um, "locate" original NES ROMs. Using PocketNES (http://www.pocketnes.org/
), you can make what amounts to a GBA ROM file and be able to play dozens of old NES games on your GBA or DS (works on both).
Also, if you're into that whole ROM "locating" thing... you might like something like an R4-DS (http://r4ds.net/
). It's like the GBA ROM cart I talk about above, but it runs DS ROMs. But I'd never ever condone using it to run commercial games. Ever. Cuz that's naughty.
There's a good amount of homebrew DS software out there too.
That's my quick and dirty DS brain dump. Hopefully there's something useful in there for you. :)