You say that the case fan runs as soon as you plug in the power cord ? I'm pretty sure this should not happen in a correctly connected system.
My guess is that you have your case connectors (front panel switches) incorrectly fitted to the motherboard.
Recheck the connections for the power switch, reset switch, and led's. Your motherboard manual should detail the correct connections.
If you didn't remove these when installing the new psu (and there would be no reason to tbh), and they appear to be connected correctly, then recheck the psu connections to the motherboard, specifically the 20-pin (or 24-pin) ATX connector and the additional 12V connector (normally a square 4-pin connector). Make sure you use the correct square 4-pin 12V connector (it should be marked 12V hopefully, and should have 2 yellow and 2 black cables).
If you were to post your motherboard model, and other hardware, then we may be able to advise better on the connections.
I see what you mean about the manual - not a great amount of detail in there.