I'd recommend these in this order
What do you like the most? Adventure, Puzzle, or Action.
Also, If you have siblings you play with, you should look into multiplayer games.
I have all of these games, so I think I can help you.
If you like action more..
1.) Super Smash Bros Wii U
-Multiplayer
-FIghting
If you are interested in fighting games, you will like this game. If you are not, you most likely will still like this game. I have a bunch of siblings and family friends and when they come over, a simple game like this can turn into a chaotic mess (of fun). Smash is a fun game, you basically beat people up, and when they get to a high percentage, you can knock them off the stage. It feels so satisfying using a charge attack and watch the character hit the screen and get launched. However, there is a competitive side to it too. So, someone can't just spam the same attack and win. This game is really simple and REALLY fun to play with friends. I think the best part to it would be the multiplayer. If you are going to get this for single player, I wouldn't reccommend it. Unless you really, really want the game.
2.) Mario Kart 8
-Action
-Racing
I would reccommend this 2nd. I don't have a burning love for this game, But it's really fun with friends.
You basically race (No way! xD) and try to make it to first place. However, there are items that stop you .
I'd get smash If I have to choose a multiplayer game, but it's up to you.
If you like adventure/puzzle more...
1.) Super Mario 3d World (I added this. I think you'd like it!
-Action
-Adventure
-Multiplayer
- Some puzzle aspects
-Only get over windwaker if you have siblings to play multiplayer with.
This game is my most favorite mario game. Never did I think I could play mario bros with friends and not bump into them every view seconds. This game is a 3d adventure game, a platformer and it's incredibly fun. The levels are so varied and there is tons of replay value because you have to unlock everything in order to gain access to the final world. This game was the reason I got the Wii U. The only think that disappoints me about this game, is that there's no online multiplayer. Nonetheless, it's fun. Up to 4 players can jump on goombas, kill stuff with fire flowers, or even suicide if they like. I recommended smash before this because I have a bias for smash, but this game is Definitely worth a shot.
2.) Legend Of Zelda Windwaker HD
-Action
-Adventure
-Puzzle
-If you are going to be playing alone
Oh wow. This game fits all your descriptions. If you are planning to play alone one the Wii U, this is the ONE game I would reccomend over any other game. I think it is beautifully remastered in HD and it's so nostalgic. I remember playing windwaker when I was little, But this brought back all those nice memories of hiding in the barrels, getting shot from a barrel, losing my sword, upgrading my bomb bag etc. If you are planning single player YOU MUST GET THIS GAME. (if you want xc) I love it so much, ad there's so many sidequests to do, I sometimes forget about the story. I've finished this game 3 times, but I feel like there's still a lot left to explore.
3.) Hyrule Warriors
-Action
- Adventure (Sorta.... Repetitive.)
I always go to my cousins house and we play this. It gets repetitive (fast) for me. If you like the dynasty warriors series though and don't mind a repetitive hack n slash, I'd go for it. But I would not recommend it.
I didn't add a puzzle section . My bad. I guess gaes fall into one category or the other. Good luck!