Hey Doc. I would like to chime in from personal experience. What I would recommend would depend on your budget, time line and performance goals. As Nonsense already said, don't throw money into the AM3+ socket or into the FX series. Neither make sense right now.
I'll outline a few situations that would determine what I would recommend to you:
1) ASAP: If you can't wait, I would recommend an i5 6400, 1x8GB of DDR4 RAM and a B or H110 motherboard. If you want to spend a little more money, get more performance and overclock a little, I would recommend an i5 6400, 2x8GB of DDR4 RAM and a Z170 motherboard in this list so you may flash an overclocking BIOS: http://overclocking.guide/intel-skylake-non-k-overclocking-bios-list/ (I can recommend the Gigabyte Z170X G1 Gaming 3).
The flashed BIOS allows for non-k overclocking of Skylake CPUs (excluding Xeons; only ASRock's E3V5 Fatal1ty C232 will overclock a Xeon at the moment). We can go more into that if you choose that route.
2) Patient Lad: If you're happy to wait, I would look at waiting until AMD Ryzen comes to. See what prices and performance are like and if Intel responds with more of their own products to compete with AMD (Skylake prices should drop if Ryzen is comparable).
As far as your GPU goes, it is on the weaker side but your CPU is going to bottleneck you in almost any game you play that is not specifically GPU bound, and I can't think of many past Tomb Raider, Crysis or CoD (all of these games will see significant improvement in minimum and average FPS, if not maximums as well, if you upgrade your CPU).
All of this information comes from the owner of an FX 4300 + GTX 660 who upgraded to a GTX 770 and saw barely any improvement. I then upgraded to an FX 8320 which still has such poor single core performance that I, once again, saw barely any improvement. When I upgraded to my i5 6400 (before overclocking), games like LoL, WoW, GTA V all have substantially higher frame rates. It removed all of the FPS drops I had to deal with. I am very happy with my upgrade, though in CAD ($) it came to over $500 for the CPU, Mobo, and RAM.
EDIT: Another note to add is the improvement in the Valley benchmark score. Almost a 10% increase (~3120 to ~3360; I'm at work so I don't have exact numbers) just because I upgraded my CPU, and this is a GPU bound benchmark. It let's my 770 show more of itself now. I know that is a synthetic benchmark blah blah, but the point still stands that even in GPU bound instances, your CPU still has to have enough oomf to help get it off the ground.