I7-3770K and gtx 970.
Csgo is solid 300fps. Skyrim with 138 2k/4k/8k mods including cities, flora and fauna, lighting, weather and still get over 60fps at ultra everything on 1080p. Metal Gear Solid V, solid 60fps with 4k DSR. Swtor capped at 90fps(or I get tearing). Guild Wars 2, over 60fps at ultra.
And yes, I own both 3570k and 3770k, similar setups and the difference is night and day in any of those games. The 3570k even has problems with some minecraft mods, simply due to lack of threads.
Granted, at 124% OC, it's a hair better than the 580, but close enough not to mention. The i5-3570k however isn't worth the investment. I5 Sandy-Bridge to i5 Ivy-Bridge only showed @5% improvement overall, so at this point its really close enough to be the same thing minus a few instruction sets. Moving from that 2500 to a 3770k would show much better results in many games. Any of the open world type games online rely heavily on thread count, the sheer amount of data from heavy server drops demand it. Also single player games like Gta5 have minimum requirements of 4 threads, but suffer heavily in fps as a result, recommended 8 thread cpus older than CoffeeLake.
$200 should be your cut off budget for upgrade. If it's less than $200, look for i7-3770K, i7 3770 or Xeon 1270v2 (if no OC mobo or just can't). Over $200 budget, start planning on amd Ryzen 2600/B450/16Gb 3200 ddr4. Somewhere around $400ish depending on sales.
If your OS isn't on an SSD, start there first. Fps might not improve much, if any, but everything else will, including load times, wait times, maps, getting around in windows etc.