Is this a sign of a failing computer power supply unit?

Whenever I do a full shutdown and try to boot up again, my case fan blows at full speed (or close to it), my hard drives spin up but do no reading or writing and my DVD drive indicator LED illuminates, showing that they have been powered on. Also, my monitor displays nothing. It just says "No input" and then after 3 seconds goes into power save mode. I know for a fact that it is not an issue with my VGA cable or with my integrated Radeon HD 3200. It will only successfully boot if I leave it set for an hour or two. I'm uming it's a PSU related issue, considering that I have a generic LiteOn 300W stock PSU. I'm really considering going out to Best Buy and purchasing a new Corsair 500W (or some other brand that I can trust), but I want to know for sure if it's going to help me fix this problem or not. I'm selling it to a friend soon and I don't want him having a sketchy computer. I also am lacking in an extra, working PSU, so I can't exactly troubleshoot it directly.