I assume that you mean a "New 3DS"/"New 3DS XL", rather than just a 3DS that is new (such a helpful name, isn't it?
)
There aren't very many games that make specific use of the New 3DS's additional functionality/power as compared to the original 3DS - off the top of my head, these are:
Monster Hunter 4 Ultimate - improved graphics
Super Smash Bros - faster loading, integration of C-stick controls
Xenoblade Chronicles 3D - N3DS exclusive
The Legend of Zelda: Majora's Mask 3D - faster loading, integration of C-stick controls
Hyrule Warriors Legends (released next month) - faster loading, integration of C-stick controls
Beyond those, though, the 3D tracking is certainly much improved and much more stable on the N3DS - but if you're looking for good value, perhaps also consider the 2DS, which lacks the 3DS functionality but can be picked up cheaply and can play all DS/3DS games except
Xenoblade Chronicles 3D.
As for whether or not to go for the XL model, that's primarily a trade-off of screensize vs portability, which comes down to personal preference, plus you have the ability to swap out 'face-plates' on the non-XL model..
Whichever model you go with, you'll be able to play all of your original DS cartridge games natively.
As for game recommendations, can you let us know what genres/types of games you'd be interested in? Regardless of your answer, though,
The Legend of Zelda: A Link Between Worlds is a mandatory purchase (
and an inexpensive one, at that).