Wii: both Zelda games, Mario Galaxy 1 and 2, Resident Evil 4 (if you haven't played any other version yet or wanna replay the game), DKC Returns, Metroid Prime 3, New Super Mario Bros., Smash Bros. and Mario Kart (if you don't wanna wait for the new games). There are other good games for Wii, but these games will already give you a lot of fun. Note that some of these games needs wiimote to play...

I think it's pointless to buy Wii games if you can't use the wiimote, you will not have the same fun without it. Try to find a way to sit at least 4ft far from the monitor. I play in a 24" monitor and sit about 4ft far from it and I have no problems at all.


Wii U: Pikmin 3, Zelda WW HD, Mario 3D World, NSMB U, Rayman Legends, ZombiiU, The Wonderful 101 and DKC:TF. There are also third party games like Assassin's Creed and CoD if you like these games.

Survival Horror:
ZombiU
The Walking Dead: Survival Instinct
Resident Evil Archives: Resident Evil 0
Resident Evil Archives: Resident Evil Remake
Resident Evil 4
Resident Evil Revelations
Cursed Mountain
Calling
Silent Hill: Shattered Memories

On-Rail Shooters:
House of the Dead 2 & 3 Return
House of the Dead Overkill
Resident Evil Umbrella Chronicles
Resident Evil Darkside Chronicles
Dead Space Extraction
Sin and Punishment Star Successor
Gunblade NY & LA Machineguns

Superhero Games:
The Amazing Spiderman Ultimate Edition
The Amazing Spiderman 2 (May release)
Batman Arkham City Armored Edition
Batman Arkham Origins
Assassin's Creed III
Assassin's Creed IV Black Flag

HackySlash or Combat:
Onechambara Bikini Zombie Slayers
No More Heroes
No More Heroes 2 Desperate Struggle
Wonderful 101
Bayonetta 2

Racing:
Mario Kart Wii
Mario Kart 8
Sonic & SEGA All-Stars Racing
Sonic & All-Stars Racing Transformed
Driver San Francisco
Need for Speed Most Wanted U
Project CARS

Fighting:
Smash Bros. Brawl
Smash Bros. WiiU
Tekken Tag Tournament 2
Tatsunoko vs. Capcom
Punch-Out

2D platformers:
Donkey Kong Country Returns
Donkey Kong Country Tropical Freeze
New Super Mario Bros. Wii
New Super Mario Bros. U
New Super Luigi U
Rayman Origins
Rayman Legends
Wario Land Shake It
Kirby's Epic Yarn
Batman: The Brave and The Bold
Ducktales: Remastered
Scribblenauts Unlimited
Scribblenauts Unmasked
A Boy and his Blob
Klonoa
Nights Into Dreams

3D platformers:
Super Mario Galaxy
Super Mario Galaxy 2
Super Mario 3D World
Sonic Colors
Sonic Lost World
Sonic Boom
Epic Mickey
Epic Mickey 2 The Power of 2
De Blob
De Blob 2

Party:
Just Dance
Just Dance 2
Just Dance 3
Just Dance 4
Just Dance 2014
Just Dance Summer Party
Mario Party 8
Mario Party 9
Wii Party
Wii Party U
WarioWare Smooth Moves
Game and Wario
Mario and Sonic at the Olympic Games
Mario and Sonic at the Winter Olympic Games
Mario and Sonic at the London 2012 Olympic Games
Mario and Sonic at the Sochi 2014 Winter Olympic Games

RPG/Adventure:
The Legend of Zelda Twilight Princess
The Legend of Zelda Skyward Sword
The Legend of Zelda The Wind Waker HD
Darksiders II
Fire Emblem Radiant Dawn
Little King's Story
Xenoblade Chronicles
The Last Story
Pandora's Tower
Monster Hunter Tri
Monster Hunter Tri Ultimate

Other:
Pikmin
Pikmin 2
Pikmin 3
Trauma Center Second Opinion
Trauma Center New Blood
Trauma Team

I was about to do what Walter did but...holy crap...yeah, that's a pretty good list. If I were to simplify it to must have games, I'd say

Wii:
-Super Mario Galaxy 1 & 2
-Donkey Kong Country Returns
-New Super Mario Bros Wii
-Kirby's Return to Dreamland, maybe also Epic Yarn
-The Legend of Zelda: Twilight Princess
-The Legend of Zelda: Skyward Sword
-No More Heroes 1 & 2
-Super Smash Bros. Brawl
-Mario Kart Wii
-Super Paper Mario, not as good as the Gamecube one, but better than the 3DS one, and still worth getting
-Fire Emblem Radiant Dawn (especially if you played the Gamecube prequel)
-All 3 operation rainfall games (Xenoblade Chronicles, The Last Story, and Pandora's Tower)
-If you never played Pikmin 1 & 2 on the Gamecube, get the Wii versions
-Metroid Prime 3, unless you've never played the Gamecube titles, in which case look for Metroid Prime Trilogy, somewhat rare, still full price, but usually not Xenoblade levels of expensive. I've seen it as low as $30

And my personal Wii recommendation:
Dokapon Kingdom. If you can actually find this game, it's usually not that expensive. Like RPG's? Ever wonder what Mario Party would be like if it were an RPG? Think battles instead of minigames, rescued towns instead of stars, and gold instead of coins, and more ways to piss off your opponents than Mario Party ever had. The ultimate friendship destroying experience.

Wii U:
-New Super Mario Bros U and New Super Luigi U (second one is significantly more challenging, if you think the NSMB series is overdone now, at least give New Super Luigi a shot)
-Super Mario 3D World, best game on the system at this point
-Nintendo Land, if you don't get it bundled, is still worth the asking price, as it has some of the best examples of unique gameplay only the Wii U can offer, particularly in multiplayer
-Wind Waker HD, if you didn't play it on Gamecube, then for the love of God get this game
-Resident Evil Revelations, if you didn't get the 3DS version, this is probably the best version to get, plays more like older RE games than 5 or 6 do.
-ZombiU, classic survival horror gameplay
-Pikmin 3, especially if you liked 1 & 2, but worth it even if you've never played those
-The Wonderful 101, a tragically overlooked game
-Donkey Kong Country Tropical Freeze, looks much better than DKC Returns and is coming out very soon.

There's a lot more on Wii U I'm looking to get, but that's all I can recommend with confidence at this point. Allegedly Rayman Legends makes some of the best use of the gamepad of any Wii U game, 1st or 3rd party, so I'll be looking into that one soon.

Wii U

Mario 3D World 

Pikmin 3 

Rayman Legends

Monster Hunter 3 Ultimate

New Super Mario Bros U

Sonic All Stars Racing Transformed

ZombiU

The Wonderful 101

Need for Speed Most Wanted U

Donkey Kong Country Tropical Freeze

Wii (Traditional controls)

Sin & Punishment: Star Successor

Xenoblade Chronicles
Monster Hunter Tri

Rayman Origins

Tatsunoko vs Capcom
Little King's Story
Kirby's Epic Yarn
New Super Mario Bros. Wii
Mario Kart Wii
No More Heroes 2 

Just a few changes to make the game harder.

And I'll list some exclusives for you:

Wii U

Monster Hunter 3U (I know, you can buy it on 3DS, but Wii U > 3DS)

Zombie U

Pikmin 3 ( a must buy for anyone that likes RTS games)

Lego City Undercover (a real fun game with greats jokes)

Sonic Lost World

The Wonderful 101

X

Bayo 2

Mario Kart 8

Metroid Prime 3, Okami, the Pikmin series, original Galaxy, Smash Bros, Wonderful 101, and DKCR would be my personal top suggestions.

Okay, New Super Luigi is a completely new game from New Super Mario Bros U. It was originally DLC for NSMBU, but they also released it as it's own game, because it literally has the same number of levels as NSMBU. The levels in NSLU are unique to NSLU and are tailored for Luigi's unique abilities. There are 82 new levels not featured in the original game. The levels are MUCH harder, and you only get 100 seconds for each.

Get ZombiU and Donkey kong tropical freeze (out on Friday)

If you are going to pick up Lost World, I also recommend Sonic Colors for Wii, it is amazing.

Pikmin 3, Tatsunoko VS Capcom, Resident Evil 4 Wii Edition, The Wonderful 101, Sonic Colors, Metroid Prime Trilogy, Skyward Sword and amny others.. There is a crapload of games you can try out.