Platformers, RPG, Shmups, Adventure
I'm going to focus on this genre by genre, just to mix things up. Though I don't have enough knowledge about shmups to make any recommendations about those.
Platformers - Both New Super Mario Bros games are very good, but I'd only recommend the second one for the 3DS. The original remains expensive to this day, while the second has more levels, better bosses, funner gimmicks, and cooperative play. The best Kirby games are probably Super Star Ultra, a DS remake of the SNES classic, and Triple Deluxe for the 3DS. Do NOT get Super Mario 64, since it lacks proper analog controls. Instead, try Super Mario 3D Land, especially if you haven't played 3D World. I also personally recommend Henry Hatsworth, an oddball game that combines action-platforming with Tetris Attack. Last, remember that the eShop has its own smaller gems, including VVVVVV and Gunman Clive. Oh, and Donkey Kong Country Returns is available and perfectly intact on the 3DS.
RPGs - This is a strong point for the DS and 3DS, provided you don't mind the games being from Japan. First, the Pokemon games. There are a LOT of these, but I would reccomend the newest ones either X/Y or AlphaSapphire/OmegaRuby. I also liked Black/White myself, and HeartGold/SoulSilver had a lot of content, but the newest games have the upper hand in presentation, online functionality, and a living community. Mario & Luigi: Bowser's Inside Story is arguably the best and funniest Mario RPG, and is full of high points and great surprises. The World Ends With You was the best original idea Square Enix put out in years, an Action-RPG set in a Low Fantasy contemporary Tokyo, and the DS version is deeper than the iOS one. Shin Megami Tensei 4 shows where Atlus learned their demon-based roots, and combines depth, difficulty, and quality like few other games. The Etrian Odyssey series is a great return to the genre's dungeon-crawling roots. The 3DS has Etrian Odyssey 4, two Untold games that remake their DS predecessors, and the crossover Persona Q. Monster Hunter 4 Ultimate has hundreds of hours of quests, Xenoblade makes me angry at Square Enix's inability to make great and expansive new Final Fantasy games, and Fire Emblem: Awakening is the culmination of what has made people celebrate the Strategy-RPG series.
Adventure - Traditional Adventure games had a bit of a renaissance of the DS, and are still at home on the 3DS. The Professor Layton series is the most famous, of which I'd recommend Diabolical Box and Unwound Future for the DS. The first entry lacked a coupl of useful tweaks, and the 3DS games made poor use of 3D IMO. The Ace Attorney series is another big series. The first three games, were sold at retail on the DS, but were later remade and are now bundled together on the eShop. Ghost Trick and the 999 games were also supposed to be great, but I haven't played them. On the more Action-Adventure side of the table, Kid Icarus: Uprising, which combines several flavors of gameplay together, is one of my favorite games. Luigi's Mansion 2 is a great game by oneself or with friends, somehow unironically being the best children's horror game I've ever played.
Zelda - I know you'll be going after Zelda regardlessw, but a brief tip about that: I'd recommend Link Between Worlds before either of the DS games. The DS games are fine, but largely focused on making good use of the touchscreen rather than reworking the basic conventions of the series. Also, Majora's Mask 3D is a more thourough update than Ocarina of Time 3D, and uses the New 3DS's second analog for camera control.
Remakes and Ports - Chrono Trigger, the BEST GAME EVER, is available for the DS. This is the best version available, lightyears ahead of the PS1 version and slightly ahead of the SNES version. Square Enix also released solid remakes of Dragon Quest 4, 5, and 6, as well as Final Fantasy 3 and 4. Star Fox 64's 3DS incarnation is very good. Last, if you like Final Fantasy, consider Theatrhythm: Curtain Call. It is a Music/RPG game with over 220 Final Fantasy songs.
