Hi there,
It will depend on the kind of games you play, but the FX 8350 can still game in a budget. More and more games are becoming multi-core/thread friendly so you should be able to stick with it for a while. Heck I am still gaming with my FX-4170 + GTX 560 TI @ 1080p! With some games I might need to lower some settings a bit but most modern games look excellent even in low graphics.
Unless you are crazy about FPS numbers going past 60 FPS or gaming at stupidly high resolution, I'd say save that money. A GPU upgrade would be the cheapest option for an upgrade though.
You also can try the Ryzen 3 1200, which comes with a beefy stock cooler and a B350 motherboard for some overclocking. There should be some money left for a cheap DDR4 RAM. The Ryzen 3 is definitely an upgrade to your FX 8350 but it's not huge at stock. You can always upgrade later to the more powerful Ryzens.
You can try the intel options but the Core i3's I found them to be more expensive to build as a whole, at least here in the US, so I went with the Pentium. It can game very well but I am not happy with it because I like the core counts on the Ryzen 3 for productivity.