Even if you could ge a 2060 for ray tracing, it isn't a good idea. Te RT cores on those first cards weren't very good and haven't aged well. They make it up with DLSS, tho.

The RX 580 that Pem has proposed is also a good option, between the 1060 and the 1070 in performance (closer to the former).

For new cards, absolutely.

For used ones, especially if they are older cards (as in Turing GTX or older), it's quite possible that their prices dropped so hard that the original price difference between AMD and NVidia has disappeared and is now more a question of supply and demand. And those NVidia's cards tend to have more supply on the used market now.

Actually the minimum requirement is I5 8400/Ryzen 1600. Apparently one of the games need at least a 6-threaded processor and Intel I5 and lower did not have at least 6 threads until the 8th generation. 

1060 6GB, or AMD equivalent, was the minimum for the game. If user benchmarks is to believe, GTX 1060 is better than RX 580, although 580 has more VRAM and the amount of VRAM may be the limiting factor for the game. The cheapest 1060 6GB sell seem to be 40€ and so is 580.

Other than power consumption, RX 6600 doesn't seem to offer much benefits overall, as the 1070, 2060 and 1070TI seem to all be cheaper and GTX1080 about 20€ more expensive than 1070TI, however, there aren't many 6600's available, perhaps it will change after the next generation of Radeon is released next month.

The new cards are expensive and even the newer used, but the 2016 and 2018 released cards' price doesn't really depend on the manufacturer anymore. As the Nvidia cards are more common, there seem to be more people who don't know what they're selling and you're more likely to find someone selling a computer with Nvidia card for less than the standalone card itself sell, than you are to find AMD card the same way. 

So this is how it's going so far: I've spent 160€ and have:
-A fractal case
-750W PSU
-Intel i7 6770K CPU
-Predator G3 motherboard
-240GB SSD
-2×8GB DDR4 2133MHz RAM
Still looking for a graphics card, most preffered cards at the moment are RTX 3050, GTX 1660ti and GTX 1070, in that order. Suitable Radeon cards are harder to find and anything above are easilly too expensive. In all likeliness the overall cost will be 210-220€. Then again, I could have bought the parts I already have 15€ cheaper and could have bought the system without graphics card and have it cheaper, but I thought the seller was in need of money, so I bought the card aswell as it might be handy.

And I found white Phantom 410 case for 10€, so I decided to buy it. I don't know if it's a hit or miss, but it does look a lot like a stormtrooper from Star Wars. On top of the extra case, I have an extra GTX 970 graphics card, Intel i5 6500 CPU and 240GB SSD. Maybe I get the graphics card I need from an age old system and put the 970 in it's place and have two computers - one on a budget and one super cheap.

For the 3050 something important is that the 8GB model will last a lot longer than the 6GB one since 6GB cards like the 1660ti are struggling in an increasing number of games so if the higher cost of that 3050 model isn't too much of an issue then it'll be the best option among those three by a big margin. The 1070 also has 8GB but it's so old that it lacks features to run certain recent games and could lose driver support within a couple years or so so both that and the 1660ti have significant issues in 2025.

If the 8GB 3050 is out of the question then you should still go with the 6GB one since it would still be an overall better option than the other two.

Quaddy will be fine.
It's not always about the threads, but clockrates, cache and IPC as well... And even TDP. I.E. Some of the early Ryzen mobile chips, the 4-core CPU's @25w would get more performance in gaming than the 6-core CPU's @ 25w because more IPC could be spent driving up the clockspeeds of the 4-core processor.

An i3 8350k can get 93% of the performance of the 8500 for example in gaming with 2 less cores.

Keep in mind he isn't running Flight Simulator or Civilization here... Even my "work" PC with an Intel N200 Quad-Core CPU can run most games "fine" if the GPU wasn't holding things back.

Keep in mind though that having just 4 cores can lead to microstutters. However, the 6770K has hyperthreading, so with 8 threads it should be fine for now.

So, now I got the graphics card. I looked for (at least) 8GB models instead of 6GB. I was too slow for one dirt cheap GTX 1080 and someone else paid more for a Radeon Vega 64 that I was willing to. So, in the end the best price per performance card that was available at a price range I was looking for, was Radeon Vega 56, Sapphire Pulse Radeon Vega 56 to be specific. Quite a power hog, but apparently undervolting can save you some energy and then again, overclocking should be somewhat possible.

Overall, the computer cost 230€, with one SSD, i5 6400 and one GTX 970 leftover. Assuming The SSD, processor and graphics card would be worth 30€ combined, it's within budget.

So, the next step would be to swap the processor and graphics card. Do I need to install drivers after I've changed them and do I need to re-install Windows? I believe the old I5 heat sink Works with the I7 aswell, so I should only need thermal paste when swapping the processor?

I'd wait to see what others with more experience than I have to say, but what I'd do is, after checking that everything works, try to find the latest BIOS for the motherboard and install it without changing the processor. Once that is done, change to processor and boot it up to make sure that it works.

Reinstalling Windows may be the best course of action, but I don't think it's required.

If the new GPU is from a different brand than the old one, like from an Nvidia card to that Vega 56, I'd suggest running DDU (link) to unninstall all Nvidia drivers and do a clean install with the AMD ones, to avoid problems.

The CPU cooler may work fine once you've applied new thermal paste, but there are software that tells you the tempereture of the CPU. If it runs too hot, you may be forced to get a better one.

So now I got the parts installed and system running. First I just simply took away the i5 processor and put the i7 in it's place, started the computer and basically everything just worked. Windows downloaded some firmware/driver update and everything was good to go. The graphics card was a different matter. The Vega card needed 2 times 8-pin power connector, but I had one 8-pin and one 6-pin, so I needed to get an adapter.

First I accidentally ordered online 8-pin extension cords that caused a few days delay and after getting them and noticing my mistake, I had to go get 6-pin to 8-pin adapters. The next problem after putting the new card in was, that the motherboard just kept beeping and the system did not start. It took an hour or two to figure out what was wrong, and so it was, that the Vega card was so much bigger, that when using the PSU's 8-pin cable, it was too tight and it put stress on some connector or something else on the motherboard that caused a loose connection somewhere - well, luckily I had the extension cord...

The system booted, I downloaded the driver update the system suggested and restarted the computer. And everything worked. The computer is noticeably faster with the i7 6700K and Vega 56 than it was with the i5 6500 and GTX 970. The games I got the system for, seem to run on the i5 and 970, despite not meeting the minimum requirements, atleast via 1080p 60Hz HDMI. Based on a quick look on hardware requirements for new games on Steam, it seems this rig will work for a couple of years, especially when the graphics card can be overclocked. It was much, much better than anything I even dared to expect. 

Overall, I was surprised how easy it was to swap the components "on the fly" on Windows. Last time I've done anything similar was in 2006 or 2007 and it was a huge pain to have anything to work. Now it was at least just as easy as it was in 2006 or 2007 on Linux.