There arent too many games that take advantage of the New 3DS over the original. Xenoblade Chronicles is exclusive, Monster Hunter 4U has improved textures and framerate, it can play SNES virtual console titles, Codename STEAM has faster enemy turn speedup. But the built in second stick and the 3D eye tracking is really nice and reasons I still want to upgrade to a New.
As for games:
Monster Hunter 4U/Generations: MH is a series that takes some time to love. There are demos on the eshop you can try out but they don't really give a full experience of the game. There is a game look up gathering materials to make potions and such, going out and hunting a monster, taking the monster parts back to craft better armor and weapons, going back out to hunt bigger monsters. The combat is real fun once you get the hang of it. 4U is reall good, Generations just released today. 3DS also has 3U but there isnt much reason to play that one at this point.
Super Mario 3D Land: Its the Mario game you should play. The system also has a New Super Mario Bros game but...eh, I've grown tired of those. 3D Land was more fun to me.
Puzzle & Dragons Z/Super Mario edition: If you like puzzle games, I highly recommend this.
Style Savvy: Run a clothing shop! Fill it with clothes to sell, customers will come in and want outfits so you dress them up and if they like it you've made a sale. It's surprisingly good.
A list, you'll know if you're interested or not:
Animal Crossing: New Leaf
Zelda: A Link Between Worlds
Kirby Triple Deluxe
Super Smash Bros
eshop games:
Pushmo
Crashmo
Stretchmo
BOXBOY
Crimson Shroud
Mighty Switch Force 1 and 2
Rhythm Heaven Megamix
Pocket Card Jockey
Various Sega 3D Classics
Hit the eshop for a good number of demos as well. Etrian Odyssey has some really leangthy demos to try out but you might also find some cool games this way.
Also keep in mind that if you haven't played since the GBA, the 3DS plays DS games just fine. Thats a HUGE catalog of games you can play and a lot of them can be found for pretty cheap now.
