You need
- 8-pin CPU cable, provided in any modern ATX PSU
- 24-pin ATX cable, provided in any modern ATX PSU
- one 6-pin and one 8-pin PCIe power connector for the graphics card, provided in almost any decent 500 watt and bigger PSU
- as many SATA power connectors as you have SATA devices (SSD's, hard disks, optical drives), preferably with a few extra for possible upgrades
That's it for connectors. If you need 4-pin Molex for something like a case fan controller or a LED strip, any power supply will have those too.
As for capacity, R9 390X is so power hungry that I would recommend a 650W unit. 550W will also work fine as long as it's high quality, but typically a 650W unit in the same model series will be quieter at load. I just don't really know what it is. I checked the website and it says its good for pc's but they were created for Bitcoin mining machines. I just don't know.
Speaking of quality:
Corsair RM650x $80 after promo code and mail-in-rebate @ newegg. 10 year warranty, excellent stability, extremely quiet fan, fully modular cables. Of course there are a little cheaper units that also fit the bill, but nothing that screams out long term value for money like this one.
EDIT Some PSU's are too big for some cases. The above is 160mm deep and will fit almost all microATX and ATX cases without problems, but if you have a particularly compact case then you'll want to make sure it will fit. For instance, Corsair 88R can fit up to 160mm but is better used with shorter units so that you can more easily connect and disconnect cables on the modular interface.