I can personally recommend Phoenix Wright, Professor Layton, and Hotel Dusk, but I don't really recommend any of them as adventure games. The adventure game elements are all pretty weak.
Professor Layton is all about its puzzles. Lots of great thinking puzzles here. I recommend all of them, in any order.
Phoenix Wright has some pretty damn good murder mysteries and it has some of the best characters and story of any game I've ever played. Play them, in order.
Hotel Dusk is the closest to being more of a functional adventure game but it's also pretty story and character oriented, and it has great art direction as well. Some of the game parts of it are frustrating though.