1) If you own the FX-8350 then remember switching to Intel (or AMD Ryzen) would mean a new motherboard, possibly system memory (DDR4 not DDR3), reinstall Windows and software etc.
2) Not sure what GPU you have either. That may be the better place to put your money if you have a weak GPU.
3) What's your gaming experience like now anyway?
If you have a good experience (i.e. High, 60FPS or so) would spending a lot of money make things better?
4) RESOLUTION has an affect on frame rate as well. The CPU is less likely to be a bottleneck at higher resolutions. You still need the CPU to be fast enough to NOT be a bottleneck (so no running 4K on a dual-core Pentium) but if you can get about 60FPS at 2560x1440 with a good graphics card then upgrading the CPU likely won't be worth the money.
5) An i7-7700K for example can increase the FPS over the FX-8350 by over 60% in some games but it varies so much it's hard to give advice. Certainly I'd start with the graphics card first.
6) Feel free to look up benchmarks for different games. For example:
http://www.gamersnexus.net/game-bench/2182-fallout-4-cpu-benchmark-huge-performance-difference
That's about the WORST CASE scenario for a slower CPU. Older game architecture that benefits more from faster cores than more of them.
But, look at THIS game:
http://www.techspot.com/review/645-tomb-raider-performance/page5.html
I doubt that's 100% accurate, but note that it at least shows no difference between the FX-8350 and good Intel CPU's. So for this game at least there's not much benefit to upgrading.
*So it varies.
(I'd show some online SHOOTERS but the benchmarks never seem to be very accurate. I think many are single player or may not be doing large client maps because there's a HUGE difference between people's results)