If you want to stick with AMD, these would be the best two pieces of hardware to get:
CPU: AMD FX-8350 4.0GHz 8-Core Processor ($149.99 @ Amazon)
Motherboard: Gigabyte GA-990FXA-UD3 ATX AM3+ Motherboard ($159.89 @ OutletPC)
Total: $309.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-11-30 22:36 EST-0500
You seem to already have a hunch that Intel would be better in the long run (it would), but the parts above will be great for playing current games and older titles with the R9 380. The motherboard is one of the best you can get for the FX-8350, and will support the overclock the CPU would require to really shine for a few more years.
You'll want a better heatsink than the stock one for the 8350, and the Cryorig H7 would be a very good choice. One of the best budget to performance air coolers available right now and fits into almost any mid-tower case:
CPU Cooler: CRYORIG H7 49.0 CFM CPU Cooler ($30.99 @ Newegg)
Total: $30.99
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-11-30 22:39 EST-0500