Here's my humble list of recommendations. I tried to not include titles that you probably have already heard of.
Bold indicates that the game has character customization.
Wii
Elebits
A cute first person shooter that has a unique "catch and throw" mechanic. Multiplayer is fun.
The Last Story
Real-time combat JRPG with a decent story.
Little King's Story - Strategy
A strategy game comparable to Pikmin in how you micro-manage your citizens to do your bidding.
No More Heroes
A beat 'em up with a cool protagonist and neat motion control combat.
No More Heroes 2
More of the last one except with more violence.
Okami
If you don't have a PS3 with the HD edition of this game, this zelda-esque adventure/puzzle game is the definitive version.
Pandora's Tower
A tense (albeit shallow plot) JRPG with third person shooter and real time combat elements.
Punch-Out!!
A boxing game that is so much more than a Wii Sports skin pack. Also slightly racist.
Rhythm Heaven Fever
One of the best rhythm games around. It doesn't rely on motion controls but rather on pressing buttons.
Sin + Punishment: Star Successor
A rail-shooter that both challenges you and itself for being over the top.
Sonic Colors
Probably the best recent Sonic game after the two weird titles that came out of Wii
Tatsunoko vs. Capcom
I dragged my brothers in this game in arcades during the summer. Really tight 2D fighter.
Trauma Team
A neat hospital setting puzzle and point-and-click where you control
first response, diagnosis, forensics, surgery, endoscopy and orthopedics that all play differently.
WarioWare: Smooth Moves
A witty compendium of mini-games that require somewhat fast reflexes to pull off. The motion controls are used well.
Zack & Wiki: Quest for Barbaros' Adventure
A cute point-and-click adventure with neat ways to incorporate motion control into its puzzles.
DS
Advance Wars: Dual Strike
Turn-based tactical game with improvements on its predecessor and an added mini game.
Chrono Trigger
The definitive version of this classic SNES title. It has extra dungeons and a new ending!
Dragon Quest IX
From the acclaimed turn based RPG series. With cool job system and character customization.
Etrian Odyssey 2 & 3
Wonderful dungeon crawling series that cuts story for pure gameplay.
Customize the characters through their classes and weapons.
Hotel Dusk: Room 215
An adventure game with an awesome story and fantastic atmosphere.
Kirby Super Star Ultra
Definitive upgrade on the SNES title. New modes and graphical upgrade. Best Kirby game imo.
The Legend of Zelda: Spirit Tracks
An easy Zelda that has the neat partner mechanic with Zelda and some great boss fights.
Mario and Luigi: Bowser's Inside Story
A solid turn based RPG with familiar characters. Cute mechanics using the brothers.
Metroid Prime: Hunters
FPS metroid game akin to the Prime series. Had a neat online multiplayer.
Nine Hours, Nine Persons, Nine Doors
A point-and-click game with a branching and solid story.
Every Phoenix Wright game
A great point-and click adventure series based around investigation and courtroom drama.
Every Professor Layton game
Puzzles, puzzles everywhere. No bad game in this series.
Radiant Historia
One of the best RPGs ever. Mixes up quick, turn-based, and grid based combat. Solid story.
Rune Factory 3
Considered by the fanbase as the best in the series.
Mixes harvest moon simulation with RPG elements.
Shin Megami Tensei: Strange Journey
First-person RPG with a demon recruiting mechanic signature to the series.
The World Ends With You
Action RPG with real-time combat. Tons of content and fashion affects combat.
3DS
Cave Story 3D (eShop)
A polished version of the legendary shooter/platformer/tearjerker.
Crashmo (eShop)
Sequel to Pushmo. You push and pull blocks in many different combinations to ge to the goal.
Etrian Odyssey 4
Most polished combat mechanics in the series. Customize classes and weapons for characters.
Kid Icarus Uprising
Rail shooter with wonderful characters and solid action segments. Customize his weapons.
In my opinion it is the best game on the 3DS but you have to get used to the controls.
Mario and Luigi: Dream Team
Another entry in the solid RPG brothers series.
Dream sequences offer cute puzzles and neat Luigi centered combat.
Pushmo (eShop)
Pushing blocks to use blocks as platforms to reach babies. Neat puzzle game.
Shin Megami Tensei: Soul Hackers
A fairly easy SMT game (saying something) and classic demon centered combat.
Shin Megami Tensei IV
Punishes you for breathing.
The first numbered sequel since Nocturne and preserves the classic series difficulty.
Tales of the Abyss
An okay port of a good Tales game. Faster loading times than the PS2 version
Theathrythm Final Fantasy
A rhythm game that features tracks from throughout the series.
Zero Escape: Virtue's Last Reward
Sequel to 999 on DS. Great story (it's practically a novel) and intricate puzzles.
Wii U
Lego City: Undercover
Basically a family friendly Grand Theft Auto with Lego wit.
Might Switch Force! Hyper Drive Edition (eShop)
A quick puzzle game whose worth comes in your ability to beat par time. Unique switch mechanic.
Monster Hunter 3: Ultimate
Practically its own genre with monster slaying and resource gathering. Difficult for newcomers.
Scribblenauts Unlimited
A platformer/puzzle game that utilizes an object creator that makes almost anything.
For your first Zelda game I recommend Windwaker HD for Wii U as it is the 3D Zelda that is most traditional in gameplay.
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