I have a GTX 1060 6GB and a Phenom II X6 1090T @ 3.7 GHz and my CPU is holding my GPU back somewhat. Since I'm using a tv as my monitor, 1080p@50 FPS is all I need, but in many games I can't max out the graphics settings and get a steady 50 FPS. For example The Fidelio Incident ran fine at 50 FPS and maximum settings in the beginning, but in the later levels I started seeing big dips into the 30s. In most first-person games I tend start out by maxing out most settings, but leaving the AA off, then adjusting if needed. Looking at 3DMark Firestrike scores, I'm about 10% off the pace that this GPU could do and in games it's those 1% and 0.1% lows where the age of the CPU starts to show. If you're not planning on upgrading the rest of our system as well, I'd say there's no sense in spending money on anything more powerful than a GTX 1060.
I took a quick look at GPU prizes on Amazon and it's pretty insane. Some GTX 970s are selling for more than GTX 1060s (3 GB). RX 470s cost 50% more and R9 390s go for double the price. I didn't do a lot of digging, so maybe these are the worst case scenarios, but it looks like a GTX 1060 3GB would be a good choice, even though your CPU will definitely be something of a bottleneck (in general, possibly not in the games you play). If you can find a good deal on a used GPU, then a GTX 970 could be a good choice as well. The 780... Maybe, if you can get it for around 100 USD or less. I would also consider getting a GTX 1050 Ti. Sure, it's not that quick compared to the GTX 1060 or GTX 970, but it's affordable and efficient. Just looking at the core count and memory bandwidth, the 780 is a lot faster but in practice games the difference is not that great in many modern games (see below), but that older architecture and all those extra resources do mean it uses a lot more power.
Here are a few links you might find interesting:
https://www.techspot.com/review/1410-gtx-970-radeon-390-years-later/
https://www.techspot.com/review/1269-nvidia-geforce-gtx-1050/
https://www.techspot.com/review/1491-top-5-worst-gpus/
(780 vs 1050 Ti, take it with a grain of salt)