^ Agree 100% when I read 380W on the PSU I really did not have to read any further. Sorry to say but the PSU is one component you just can't skimp out on. Return that PSU if you can and get one of the one's that andrewcarr suggested. Plus as andrewcarr said you HAVE to have both PCI-E power connectors plugged in or your rig will not work. With a AMD-FX 8150 and a Sapphire HD 6950 I would not go lower than a good quality 500w PSU.
Other than the PSU I think you got a good rig there. Get a good PSU then double check all your work and when you are 100% everything is good power it on and you should be good to go. Do not get to upset as this sounds like your first build. Mistakes will happen it is all part of learning. I bet almost every one on this board has made a newbie mistake at one time or another it is part of learning and carrying that info with you on the next one. Good luck and I really hope you enjoy it.
Or do I have to buy a new power supply? Kinda hard pressed for cash ATM. I'm definately planning to get a 500w either way though.
I also just used the paperclip test to test my PSU and its working.
Is there any way to test my other components? I want to know if I fried my motherboard or something. Shouldn't the fans be running, or shouldn't it be making a beeping noise? Literally nothing is running.
Yes you should still be able to return the PSU, depending on where you bought it though there could be a restocking fee.
As for the motherboard I don't know of a way unless you can get another GPU to test/setup the system on (that supports the GPU). Since the motherboard doesn't have any built in video you can't run the computer off of that until the new PSU comes. I say new GPU because if you just tested it with a low power one and the PSU you have it should post.