What games would you recommend to someone who just made the switch to a proper gaming PC?

I'm not sure how helpful it is, but here's the list of games I own, but can't run on my integrated graphics without bleeding eyes:

View attached image.

Just use lists to find interesting games. There are plenty of these, covering all kinds of aspects.
Opencritic - highest rated PC games
Steamdb - highest rated
Steamdb - most followed
Or just check https://steamdb.info/ for most played, trending etc.

Often you can use filters to find games for your fav genres etc.

DOOM Eternal
Sekiro
No Mans Sky

For the best possible story in gaming history: RDR2 - Yakuza 0 - 6 - Witcher 3 and NieR Replicant / NieR Automata. Those games are pure emotional mastepieces.

I can't tell ya how much i love yakuza and NieR. Never experienced anything like that.

Edit: totally forgot about Half Life Black Mesa , 10/10 if you ever played Half Life 1 or if you want to experience Half Life for the first time.
Edit 2: Forgot also about Days Gone, just played it recently and had a blast.
Edit 3: DYING LIGHT if you love zombie games and parkour. <3

Control
It's the best Jedi simulator (and one of the best action games) I've played. And it's dirt cheap on sales on Steam.

What kind of games are you looking for? The list is long without any precision from you ^^

if you love RTS tactical gameplay do not miss out on
commandos 2 (or whole commandos series except commandos strike force)
Desperados: Wanted Dead or Alive
Desparados 3
Shadow Tactics: Blades of the Shogun

Honestly no idea what type of games you are looking for, but I'll settle with games I enjoyed a lot with awesome visuals as well.

Batman Arkham Knight
The Witcher 3
Cyberpunk 2077
Doom (2016)
Hellblade
Tom Clancy's The Division 1
Divinity: Original Sin (both)
Prey (2017)
Rise of the Tomb Raider
Ryse: Son of Rome

If you're into graphics-heavy recent beautiful games... I guess playing Monster Hunter World was really impressive for me the first time I booted it up.
Control has impressive scenario as well... Maybe Death Stranding too, it has massive open spaces and wilderness and a lot of detail. Outer Worlds is a good bet as well, if you get your FOV very high and everything on the best graphics setting.

Control.

Yakuza! I suggest Kiwami 1 before the unmitigated masterpiece that is Yakuza 0, because I was so invested in learning the backstories of characters I was already attached to that I firmly believe its the better order. 0 is first chronologically, but as the number suggests, its a prequel. Then if you want more, play Kiwami 2 and then the remasters in order.

Earth Defense Force 5

CyberPunk , Darkest dungeon , Counterstrike Golbal offensive,, Stellaris, Stardew Valley Terarria Bioschock infinite, GTA V, Farcry 5. Control Euro Truck Sim 2 .