The R7 370 is just a faster version of the HD 7850. The GTX 670 is a substantial upgrade in performance from that, but it uses more power. I wouldn't bother with the GTX 680. In new cards, the 1050 Ti would a bit above the 670, plus you get 4gb vram, plus it requires very little power compared to the other cards. If it were me, I'd be tempted to get an RX 470 if that's available and in your budget.
It depends on your power supply. If you already have enough power for all these cards, then it's just a matter of choice. If you will need to buy a power supply, then you'll have less to spend and therefore not as much choice.