First step: I suggest trying a known-good power supply if you can.
Caution - the large capacitors in these get charged to hundreds of
volts, high enough to be dangerous. I've seen a few power supplies fail
for reasons other than the capacitors, and when the caps fail they can
damage other parts in a chain reaction. Computer power supplies are not
designed to be repaired IMO.
[...] I never took a power supply apart before. Anything special I
should watch out for?
Generally, there are a couple large capacitors that are cooked. The
cans look a little "overinflated". They aren't hard to change - but
in reality it isn't worth doing. A good replacement supply is $50 or
so. If a power supply is questionable at all it isn't worth taking a
chance.
The caps can retain power for a while, so shorting them can be ...
shocking.
-- Bill