-Fire Emblem: Awakening (easily the best game on the 3DS imo)
-A Link Between Worlds
-Pokemon Omega Ruby/Alpha Sapphire (actually get all pokemon)
-Pokemon Mystery Dungeon: Explorers of Sky (criminally underrated game)
-Ocarina of Time 3D
-Super Mario 3D Land
-Mario Kart 7
-Mario and Luigi: Bowser's Inside Story
-Kid Icarus: Uprising
-Bravely Default

Most people would post the same games I have in mind to mention, so let me just suggest that you buy mainly games that can't be played on home consoles, or won't be better on home consoles. For example, games where the camera has a bird's eye view (like every handheld Zelda besides the N64 remakes, or GTA Chinatown Wars), games like Ninja Gaiden DS or Trauma Center where the gameplay is heavily or totally reliant on the touch screen, and Pokemon games.

There are many games but the game i would suggest the most is New Super Mario Bros (DS). Its my fav game of all time due to the amazing CO-OP. The single player is great too.

3DS
Xenoblade 3D
Super Mario 3D Land
Super Smash Bros for 3DS
Fire Emblem Awakening
Mario Kart 7
Zelda: Majora's Mask 3D
Kirby Triple Deluxe
Animal Crossing New Leaf

DS
Mario Kart DS
Pokemon Platinum
Wario Ware: Touched
Zelda: Spirit Tracks
New Super Mario Bros
Mario and Luigi: Bowser's Inside Story
Sonic Rush
Kirby Canvas Curse

I'd recommend the Ace Attorney series (there's a remastered version of the original trilogy that's on 3DS), Fire Emblem: Awakening, Resident Evil Revelations (if you don't have a console that can play it), and Ghost Trick.

Yeah now it is easy since you will not have to worry about Animal Crossing or NSMB or Mario kart twice so my list is:

3DS:
In no order...

Kid Icarus Uprising
Super Mario 3D Land
Super Mario Kart 7
Resident Evil Revelation
Animal Crossing New Leaf
New Super Mario Bros 2
Super Smash Bros
Luigi Mansion 2
Pomen x/y
Monster Hunter 4

DS:
Mario and Luigi both games
The World Ends With You
Super Princess Peach (so cute)
Final Fantasy III/IV
Okamiden
Professor Layton - you pick one
Dragon Quest IX: Sentinels of the Starry Sky
Golden Sun: Dark Dawn
Pokemon Black/White

Super Mario 3D Land
Ocarina 3D
Majora's Mask
Luigi's Mansion


And I recommend all Professor Layton games as well, DS and 3DS, and Pokemon games (the remake of Silver and Gold being the top of my list)

Thanks for all the suggestions! Got a good hunch on the games by now, my first buy will probably be Bravely Default.

Also a question on Xenoblade for the 3DS.. are there any differences from the original?

There is literally tens of thousands of hours worth of fabulous games you have missed out on by not buying these games lol! They are literally my childhood. Here are the important ones: one Pokemon game of each generation( I recommend platinum, HGSS, Black 2 White 2, X or Y, and ORAS), Mario and luigi bowsers inside story and dream team, both Mario Karts, Pokemon mystery dungeon explorers of sky ( This is a MUST HAVE ITEM NO EXCEPTIONS) and pretty much everything else everyone else has mention thus far. Good luck with your game hunting!!

Most people already recommended my favourite games for the systems, so I'm just going to leave a little recommendation here that I haven't seen mentioned much so far:

The Ace Attorney series is very good and the english localization is super funny (in the best way possible). The original triliogy has been remastered and is available from the e-shop for relatively cheap, so is Dual Destinies the newest entry in the series.

If you're into puzzeling, georgeous visuals and soundtrack as well as wacky stories and characters the Professor Layton series might be for you. It a mix of georgeous point and click adventure, visual novel and brain teaser puzzeling, all in a very charming package. There are six games and you can probably get the Ds ones used pretty cheaply.

As a little bonus, there even is an Ace Attorney/Professor Layton crossover game.