This may be one of those grey areas that may be hard to diagnose if you don't have spare parts you can swap in, to narrow down what could be wrong. First I would see if those lights on your MB are diagnostic lights, maybe when they are lit they are telling you something.
Second, I would disconnect everything but the bare minimum to get to the BIOS... no hard drive, no peripherals (not even a mouse), no CDs or floppies if you use one, just keyboard and monitor. The idea here is to use as little juice as possible, so that if it is a power supply problem, you're stressing it as little as possible. If it works, leave it running for a while and see if it shuts down again. If it still works, add things back in order of importance (mouse and hard drive before CD drive, before peripherals) until it re-fails. That should tell you something, either a component that is broken and drawing too much power, or that your power supply is dying and isn't giving enough power.
Third, I don't know what a power supply tester is either, but I don't think most people can be expected to just have one. I just googled it and this was the top result, I agree with what they say:
http://pcsupport.about.com/od/toolsofthetrade/f/powersupplytest.htm .
