POWER SUPPLY REPLACEMENT

Hello,
I have an older model hp desktop that I installed a new ssd drive in along with my old drive, it automatically kept shutting down, so I took out the old hard drive and just put in the ssd, it continued to shut down, so I took out the ssd drive and put the old hard drive back in and it works fine, the power supply is only 350 watt so from reading i figure that is my problem, I just need to know what specifics I need to look for when buying a higher watt one. is 600 watt too much, is there a limit to what I should use, what other specifics do I need to look for.the old one is a Hipro HP-D3057F3R hp p/n 5188-2625. any help would be appreciated.