In my opinion the most essential Wii U games are:

Mario Kart 8
Smash Bros
Splatoon
Bayonetta 2
Donkey Kong
Pikmin 3
Super Mario 3D world
Zelda Wind Waker

Deus Ex Human Revolution is also really good if you've never played it before. I own both 360 and Wii u version, and Wii U version is better.

I also reccomend getting a Wii Remote + nunchuck. Work with most Wii U games for multiplayer, and allows you to play Wii games.

Both Mario Galaxy games, Punch out, Metroid prime trilogy (downloadable on Wii U!), Skyward Sword, Sin and Punishment, Little Kings Story are a couple of Wii games I'd reccomend (although there are more of course)

Edit: pro controllers works with most multiplayer games like Smash and Mario Kart, although there probably are a few exceptions. But it's a great controller with amazing battery life (80 hours)

Mario Kart 8, Super Smash Bros, and Splatoon.

Pro controller is nice to have, but keep in mind that some games don't support it.

If you're interested in using GameCube controllers with Smash, you can get an adapter from Nintendo's online store.

That is amazing news! So I could get Metroid Prime trilogy today? Can i buy additional memory or something?

Nintendo generally sucks with digital. It's good for indies but their 1st party games are nearly always cheaper in physical.

No, Splatoon doesn't have splitscreen. Either 1 vs 1 on TV and gamepad or online multiplayer.

Online store - eShop is good. check out the sales event where good discounts are there and good VC games coming in.  there are a few demos as well, so check it out. 

If the price for digital and physical is the same, go for the physical copy.  take note that the disk space of a Wii U deluxe is just 32gb, so you need to buy an external harddrive or USB thumb drive(haven't tried the USB thumb drive though) which is easy to setup.

With regard to Splatoon - no there is no splitscreen mode for Online mode (Turf Wars and Ranked Battle).  It has a 1v1 local, which the 1st player uses the gamepad and the 2nd player would use a pro controller.  Not sure if a WiiMote can be used for this mode.

What genre do you like? that would be helpful on the games you may want. 

That kind of sucks, but I will just have to get most games physically then, no big deal :).

just use an usb stick or a hdd. no problem.

First, I'm going to link to a thread from about a year ago, just before Mario Kart 8 was released. It was the Top 11 Wii U Games as chosen by the VGChartz community. It's outdated by now of course, but the 11 games listed are all still great for the reasons mentioned in that thread.

Besides those 11:

• Mario Kart 8 - Like Smash Bros for Wii U, it is arguably the pinnacle of the series, regardless of what certain Double Dash lovers have to say. MK8's best strength was that it seemed like the logical culmination of the series, drawing the best elements from the Wii and 3DS iterations and ironing out the flaws in the process. Stuff like the item-fueled chaos of the Wii games was toned down, the content was far more expansive than in MK7, and even the seemingly gimmicky anti-gravity meshed into existing gameplay very well. Online play is also very good, lack of voice chat aside. It may not be worthwhile for the solo player, but either local or online play make it worthwhile.
• Super Smash Bros for Wii U - Almost the perfect Smash experience. The roster somehow manages to gain a dozen new interesting fighters, the entire roster is better balanced and truer to their original incarnations when applicable, and the items, stages, songs, trophies, and so on are ridiculously numerous. Though it lacks the single player Subspace Emissary of Brawl, SSB4 comes ahead in every other department. Playing with friends is amazing thanks to 8-player Smash, customizable Mii Fighters, and the stupid but enjoyable Smash Tour mode. Online play is lightyears beyond Brawl, including two distinct components for casual and competitive play. The Events and Orders modes give a solo player plenty of worthwhile challenges to undertake.
• Splatoon - Nintendo's best multiplayer shooter since the N64. The emphasis on territory over kills gives this one unique dynamics. It lacked a bit of content at launch, but already has largely made up for it with free updates, which are rolling out on a regular basis.
• Bayonetta 2 - Did you like any of the Devil May Cry games? Or any modern Japanese action game? Then get this, and free updated version of the original game while you're at it.
• Hyrule Warriors - Come for the Zelda references, stay for the 100+ hours of content and over double that with some great DLC.

As for the Pro Controller, it is not used for all games, but is still widely used. My friends use it for Smash Bros, Mario Kart, Hyrule Warriors and Super Mario 3D World when we play together. One particular friend was really into Hyrule Warriors for a couple of months, and the Pro Controller made that game MUCH better.

That nursery stage definitely scared the crap out of me.

While not the prettiest game ever made, ZombiU certainly was a fantastic horror game for me.

Wii U is compatible with external hard drives

But the game fits on the 32GB Wii U. It's only 7,7GB despite being 3 very lengthy games ( them being no HD of course)

But yes, you can get it today. Best part is that it's only 20 dollars. (requires Remote + nunchuck btw)