Oh man, there is TONS of good stuff on the DS in both RPGs and Adventures/Visual Novels.
First off, I have to link our DS reviews page: http://www.rpgfan.com/reviews-ds.html
Secondly, I have to suggest the following:Visual Novels
Phoenix Wright 1-3 (and Apollo Justice) - These are on the wacky side, but are some of the best visual novels/adventures available. They're tense and heartwarming and all sorts of other emotion-y things.
Hotel Dusk: Room 215 - This one falls on the more serious side of the adventure genre, but it's a great title from the now-defunct Cing. There's a sequel, The Last Window, which was only released in Europe. Still, the original DS is region free.
Lux-Pain - OK, Lux-Pain is niche even for an adventure game. It's a science-fiction adventure... and its localization is one of the worst I've ever seen. There's engrish and typos everywhere. But what's behind it is really quite good.
Professor Layton & (Insert crap here) - these are much more focused on puzzles than the story, but they're all quite good as long as you don't expect a masterpieceRPGs
Radiant Historia - This is pretty much the best RPG on the DS. Reminiscent of Chrono Trigger w/ its timeline/time travel business.
Might & Magic: Clash of Heroes - if you have a PC or Xbox 360, those versions are better balanced, but the original is still a great puzzle/RPG hybrid.
The World Ends With You - Square Enix's sleeper hit, this is one RPG that does a lot with the DS hardware. Its story is hit-or-miss, but if you like it, you'll REALLY like it.
Dragon Quests IV - VI, Chrono Trigger - Remakes of Super Nintendo Games. Not much needs to be said about these.