The market is a mess right now but the basic rules still apply:
Devote a great deal of your budget to a good GPU, decent CPU, and quality PSU. The rest can be cheap.
I generally recommend people take a moment or two to browse eBay for a CPU+mobo combo from yesteryear; generally Intel 4th gen stuff is accessible and budget friendly. If you’re entirely against buying used components than AMD’s Ryzen 3 parts are a godsend as they dip to around $100 for a quad-core (no SMT) and $50 for the barest of barebone A320 motherboard. The upgrade path for AMD’s AM4 platform is great as AMD have already stated it will remain in use for awhile, possibly past 7nm Zen2.
So here’s a list of all new parts I’ve put together that nearly meets your budget. It’s not fancy but it’s going to work for your purposes today, and has drop in upgrades for tomorrow: https://pcpartpicker.com/list/RDWsHN
Now here’s what you can get on the used market:
Intel Core i5-4570 - $109
HP Q87 mATX Motherboard - $25
8GB Hynix DDR3-1333 - $40
GeForce GTX 580 - $60
So in a more comprehensive price list it’s going to look a little like this: https://pcpartpicker.com/list/TQdQ3F
(Obviously PCPartPicker doesn’t have some of these items, so substitutes are used with the proper price attributed.)
Don’t take either of these as recommendations. The only thing I recommend at this point in time is to increase your budget beyond $500 before looking at building a new system, and keep your eyes on eBay for last-gen GPUs as anything modern is being absorbed into mining rigs. GTX 670s are a decent deal around $100, and the GTX 580 is a fantastic deal under $60. Those are still fully capable of 1080p gaming even today.