I won a DS a little while ago and was really taken by it. It's probably the most fun i've ever had out of a handheld, because the gaming is so innovative and new, and doesn't rely per se on the old push button make move gimicks. Using only the touch screen in some games is terrific.
Unfortunately, I find it severly lacking in good games to play.

I have at the moment Animal Crossing Wild World, which is basically a touch-screen rehash of the GC version, with some extras. It's fun, but gets old quick.
The New Super Mario Bros. This game is AWESOME if you're a fan of Super Mario 1 and 3. It basically takes all the best elements of both of those games and combines them together with Mario64 graphics and moves, and gives you a long list of new levels to play through. Love it.
Mario64. A remake of the same game for the Nintendo 64, with all new characters and secrets and updated graphics etc, extremely fun.
Resident Evil DS, a remake of the very original RE game from the PS1, with some updated graphics and a bunch of nifty new things that take advantage of the touch screen.
I also want to pick up Mario Kart DS, naturally. I gave FF3 a go, but got bored of it relatively quickly. There's only so much oldschool FF I can play.
Coming out soon are two new games i'm extremely excited for: Touch the Dead, a zombie-shooter on rails similar to Segas House of the Dead, and a direct sequel to Windwaker, which looks like an excellent cross between Zelda: link to the Past and Zelda Windwaker, a great combination.