Bravely Default: An action JRPG about saving the world with heroes by activating crystals. (a sequel for this game, Bravely Second is coming out next year sooo get this amazing game and catch up to the series :p)
Fire Emblem Awakening: An SRPG about war. If you do get this game, play it on Classic mode and hard. Trust me, it's fun that way. Also a new game, Fire Emblem Fates is coming out next year so you should also get that.
Zero Escape This is a sequel to the DS visual novel, 999. You should try it.
All the Zelda games are worth trying as well, along with Pokemon. However the three games in bold are the 3DS must haves.
You should also try Kingdom Hearts: Dream Drop Distance, and Animal Crossing New Leaf.
If you got a New 3DS, get Xenoblade Chronicles 3D. It's a really fun RPG game about saving the world from robots. I wouldn't say the story is good, but I just love walking around on the massive open world of that game. But if you got a Wii U or a Wii, get this game on there instead because the game was originally a home console game.