I just picked up a DS, and I must say, it's a very nice little machine. Ultra-portable, long battery life, built-in touch screen, microphone, and wireless, and a solid lineup of titles including a long history of GameBoy Advance titles which are still compatible with the machine. (Titles for older Game Boy models, such as original GB and GBC carts, will not work with the DS - only GBA titles.) This is particularly nice as I already have a small collection of games for my aging GBA SP, some of which I've now picked up and started playing again.
I've had it about 48 hours, and so far, I'm extremely happy with it. Final Fantasy III for the DS is a great game in the classic Final Fantasy tradition. Mario Kart DS is a solid title in the Mario Kart line, and blows away any portable Mario Kart title to date - honestly, you'll be amazed at the quality of the graphics and gameplay packed into a handheld game.
Elite Beat Agents is a good game for fans of rhythm games (which I am, even though I tend to be terrible at them). It uses the touch screen to great effect and provides a good challenge throughout, with as much replay value as can be packed into the relatively few songs included with the game.
I have yet to play multiplayer (DS-to-DS or via Nintendo WiFi Connection), but if you'd like to help me give it a shot, post your friend codes in comments.
My Mario Kart DS FC: 274976-668344
Castlevania: Portrait of Ruin FC: 090293-173479