RedEye, Flavrans and CPU have given you good advice on how to spend a couple of hundred dollars to eke some more life out of that ancient motherboard, however with your stated budget of around $600 I absolutely wouldn’t bother. By ripping it out and spending your full budget you’ll definitely get a far better system that stays capable of running modern games for a lot longer and has room to do incremental upgrades in future years.
Looking solely at Newegg (sorry for lack of links but I’m doing this hastily at work!) here’s a system built on a six-core mid-range Ryzen CPU, 8GB of nice speedy memory (at the point just before faster RAM starts to get rapidly more expensive and shakily supported by all but the fanciest motherboards), and a very reasonable price on a capable mid-range AMD GPU that will run most games at Ultra detail at 1080P resolution and reasonable frame-rates. All on a well-specced motherboard that can easily take more or faster RAM, and CPUs and GPUs of the next generation or two if a couple of years down the line you have a few more hundred to spend and want to keep up with the latest games or a higher-res monitor.
X370 Motherboard - $90
AMD Ryzen R5 2600 - $170
8GB DDR4 3000 RAM - $56
AMD Radeon RX580 8GB - $200
Western Digital Black 1TB - $73 (I’m stealing RedEye’s suggestion - keep the SSD as a boot drive and get a nice big HDD for games and data. You can always replace the SSD later.)
Total price = $589
All that assumes you have a decent PSU; if you need one might as well get a good one from a known brand, well-rated for power efficiency with modular cables and room to expand because it should last you many years:
600W 80PLUS GOLD certified from big brand like EVGA - $70.
If that takes you over budget, drop the Ryzen R5 2600 (which is a little overkill for this machine right now) down to a four-core low end R5 or an R3 (like the 2200), to offset the cost of the PSU and keep you under or around $600.
Good upgrades in the next couple of years are another matching 8GB of RAM to take you up to 16GB, that’s relatively inexpensive, and if you do drop the CPU down to a cheaper one to afford a power supply or something else you could look at getting a beefier one from next year’s 3000 series Ryzens, or the generation after that (should all still work in that MB with a BIOS update). For a more responsive PC in general you could keep an eye out for good deals on M.2/NVMe SSDs to replace the current one as your boot drive, with more space so you could put your favourite or most played games on it and get faster load times. As long as you’re just using an old (presumably 1080P/HD) TV I wouldn’t worry about upgrading the GPU. If a few years down the line you get a 1440P or 2160P (4k/UHD) monitor then you’ll want to save up and put some cash into whatever the best GPU you can afford at the time is.
(If you have absolutely no interest in upgrading this machine again we can probably readjust some of the spending away from flexibility and future-proofing and invest in a little more bang for buck.)