Thanks all for the suggestions. Too many to reply individually, but I will look at all of the games mentioned (trailers, Youtube playthroughs etc.) 

Currently leaning towards Witcher, Doom, and some of the strategy games suggested.

I've bought a new PC to actually be able to play modern PC games (specifically Yooka Laylee and Talos Principle). 

I would like to buy a game that can use the full spec (GTX1060, 1080p).

Are there any recommendations for AAA releases that push graphics and are well made games? I know absolutely nothing about AAA PC releases. I have no preconcieved ideas about which series to play or not play.

I have mostly played Nintendo games up to now. I have zero experience of shooters or shooter controls. I am innately bad at fast-paced games. I like games with strategic depth, but not high mechanical intensity. I'm willing to try a shooter if there's a steady learning curve. 

If you're asking for exclusives there aren't many lol
Not games that push the hardware (apart from the Total War series, but which I don't like)

But AAA multiplatforms are plenty that look amazing on PC.

I recommend Dark Souls 3, Fallout 4, The Witcher 3, Doom, Lords of the Fallen and possibly Assassin's Creed Syndicate. Graphically demanding and beautiful games. And real good games. My primary choice of those, for you, would be Fallout 4 because it offers so much. Visuals, depth, tinkering, variety, RPG, exploration, great shooting mechanics but not over the top and not too demanding. If you don't like Fallout 4 as a newcomer to PC games coming without any preconceptions you might as well sell your PC right away lol

EDIT: not as graphically impressive, but you have to check out XCOM 2. A thinking man's game, yet modern and AAA, and doesn't demand any reflexes whatsoever. I adore XCOM.

Gratz BTW.`Does it feel good? What specs does your new comp have?

Deus Ex with HD texture pack

How open are you to modding?

I used to be kind of an anti-First Person elitist but then I tried Fallout 3 and discovered that many FPS-RPGs can be played at a slower, more tactical pace with stealth and sniper builds. Even series like Borderlands allow you to play mostly as you like.

I'd recommend playing Dishonored in preparation for Dishonored 2.

Some possibilities (with not all being all that recent or pushing the graphics very hard, but AAA nonetheless):

• Grand Theft Auto V
• Total War: Warhammer (or some of the previous games)
• Battlefield 1 (once it comes out; could be unforgiving for a shooter noob though)
• ArmA III (I would imagine it's a bit too hardcore as a shooter though)
• Deux Ex: Mankind Divided
• Civilization VI (won't be pushing graphics but it's an AAA game; not yet released though)
• The Witcher III
• Cities: Skylines (not that intensive for your PC, but you could probably push the boundaries with some mods; also, more like AA than AAA)
• Metal Gear Solid V: The Phantom Pain

I'm not sure how well they fit into your requirements, but I tried to give them some though about it. I'm mainly throwing around some ideas you might want to look into further to know if you're interested. Disclaimer: I haven't played all of them, but I imagine I could like in quite a few of those (actually all of them, to an extent).

Skyrim isn't exactly a new thing, but you could try it too with mods. With mods, it can still probably test your hardware. There's a remastered version coming soon, and you'll get it for free if you own the Legendary Edition of the original game.

Need not be exclusive.

i5 6600K/GTX 1060/16GB RAM. I could have spent more on graphics but I won't go above 1080p so it's not worth it.

Would watching  a random youtube playthrough of those games give a good impression of how they play? Fallout and Witcher seem the interesting to me. Souls is probably too hard. 

Modding is OK but I'd want a reasonable standard of quality. 40 mods conflicting with each other and a jank UI with 200 toolbars is not my thing.

Dishonored is a stealth game, right?

Forza Horizon 3 is just out and supposed to be one of the best racing games ever, I would buy that
Almost time for Gears of War 4 (should be great)

Some older titles, Ryse will look nice to test your graphics card but its not an amazing game.
The Total war games are great if you like strategy and really give a scence of battle and amazing graphics.

Civilization is one of the best reasons to own a PC

And don't forget to play hearthstone.

have fun with your PC, PC is the best :P

Did you miss The Witcher 3?

How about Middle-earth: Shadow of Mordor or the amazing Dragon Age: Inquisition?

You can't go wrong with any of those 3. Fantasy games play amazingly well with keyboard and mouse.