Kirby: Canvas Curse is, if you can manage to find it, a MUST. The game is simply awesome, and it was one of the first games that made me go "WOOH DS :D". Very easy to pick up and play, but the mechanics are fun throughout the game, and it looks gorgeous as well. This game is just FUN.
I'd also recommend Meteos (again, if you can find it). Its a falling-block game, somewhat similar to Tetris but not really, actually. I can't vouch for the "disney"-themed version of the game, but the original is excellent in every way except for its lack of Online Multiplayer, and once purchased stayed cemented within my DS until I had purchased Mario Kart DS.
Speaking of Mario Kart DS, buy it. Its the best Mario Kart ever. Seriously. I haven't played Diddy Kong Racing DS, but from what I understand Mario Kart is a superior product, and I know for a fact that more people have Mario Kart DS, so it would probably be more worth your money. I see no reason to buy both.
New Super Mario Bros. is good, but also easy. You'd probably get more bang for your buck downloading an older Mario game off Virtual Console, but the game is undeniably well made and has three special coins for you to hunt down in each stage, as well as mixing in some of the 3D Mario gameplay with the 2D. Really, the only thing wrong with this game is the fact that its so damn easy, and if you like Mario games its highly recommended.
I'd be wary of Animal Crossing and Nintendogs. While they are both excellent games (and I love them both), they both suffer from the "never ends" aspect, which is both a blessing and a curse.