i would suggest to go with either a 8GB RX480 or 6GB GTX1060 if possible.
And if you could go with the i5-6500 instead of the i3.
The i5 is a true quadcore, and for todays modern AAA games it realy will be a benefit.
i5-6500 + RX480 / GTX1060 is basicly a rock sollid 1080p gaming combination.
But yeah with a $500,- tops this will probably going to be hard.
Allthough a RX480 4GB card wont be that bad either.
Ok this video is basically made for your case:
i would suggest the sapphire 480. the 1060 and 480 trade blows pretty well but the 3g of vram hurts. sapphire version is the best of the aftermarket 480's and it's pretty well priced too. on the 1060 side you really should go 6gb version if you get it and dont get stock/founders. several modern games are using 6+ gb's at 1080p these days and this is going to continue in the coming months and years. a 3gb limit is going to force you to upgrade much sooner.
also for the 480's stay away from the gigabyte and devil. the gigabyte gets really hot and the devil has significantly worst vrms then even stock.
I hope this isn't in reference to gaming performance... there's a negligible difference between 2.0 and 3.0 or even x8 vs x16 read this if you don't believe me: https://www.pugetsystems.com/labs/articles/Impact-of-PCI-E-Speed-on-Gaming-Performance-518/
It's what I mean, if you bother to read the other half of that sentence you just quoted the first half of:
I'll say go 480 4gb or if you can swing it get the 8gb version. It's better in dx12 and vulkan, will be better in dx11 through driver improvements (happens pretty much every generation), and if you want to get into the "sync" monitors freesync is a minimum of $100 for a same spec gsync monitor but the difference is usually 2 or more times that.
Common complaints about AMD:
"it runs hotter" So what? If you're like me and live in a cold place you like the extra heat in winter and you can save a little $$ on your heating bill ;) (not to mention that it typically has no/negligible effect on performance)
"it draws more power"-
the gigabyte 480 has issues
I recommend neither for various reasons.
Concerning the AMD heat issue: When talking about Zen, we don't actually know how hot it is going to run, so we just have to wait. However, AMD GPUs run hotter, but most custom solutions have a decent cooler and with some undervolting, you will be able to reduce the temperature as well.