1. Unless you know japanese perfectly, don't buy DSi (DSWare region locked, menu fully in japanese), won't hit US until "well in 2009".
2. Original DS and DS Lite - yes, DSi - no.
3. RPG - The World Ends with You, Chrono Trigger if you didn't play it back in those days or FFTA2: Grimoire of the Rift if you're fan of tactical role-playing games; Action-adventure - The Legend of Zelda: Phantom Hourglass, Ninja Gaiden: Dragon Sword if you like hack'n'slash or any Castlevania; Platformers - New Super Mario Bros. or again any Castlevania if you consider them platformers +Yoshi's Island DS - this one my own favourite, but honestly speaking just a rip-off of original SMW2: Yoshi's Island on SNES, which one is a great game though; Shooters - Space Invaders Extreme, if you've already played this one on PSP then Bangai-O Spirits (from FPS genre Metroid Prime: Hunters only worth mentioning); Fighters - nothing worth playing, imo (japan-only Jump Ultimate Stars quite a fun though) +for turn-based strategy fans - Advance Wars series must play, surprisingly very addictive gameplay for it's genre +don't miss adventure\visual novel genre on DS, which pretty much not-existant on PSP, try Ace Attorney series, Trauma Center series, Hotel Dusk: Room 215 +puzzle genre like Professor Layton series or a bunch of addictive casual games like Rhythm Heaven (Rhythm Tengoku in Japan).