I know Im late to the game, but Up until the Intel Core2Duo CPUs I was firmly in the AMD camp. The performance gulf between the C2Ds and AMDs multicore CPUs is impressive - to the point of being well worth the extra investment.
As for cooling solutions, you would be best to go 3rd party regardless if you choose AMD or Intel, unless you are going to use your PC mainly for business applications and web browsing, then the stock coolers should be fine
Ram: Go for at least Dual-Channel DDR2, at least 2 gigs worth, especially if you are running Vista. Vista is a resource pig.
Graphics: Either an Nvidia or ATI solution - If you are looking for game playing, choose Nvidia, if you are looking for Video/DVD etc, ATI should be your choice. Don't rely on integrated graphics solutions, regardless of who makes the preinstalled system. The integrated graphics perform poorly by comparison, and they rob your system of some of your installed ram, which helps to decrease performance systemwide.
Integrated sound is just fine - it doesn't have the same resource crippling effects.
Mainboard - Im a fan of stuff by Gigabyte, Asus, or MSI - I don't care for most other makers, especially PC Chips. Avoid, avoid, avoid.
Power supply - get something that puts out at least 500 watts. much less will result in system strain, which leads to overheating which usually results in a shortened life or failure.
I'm ambivalent on anything that you connect externally to the system, so get whatever mouse/keyboard/monitor/printer that you feel will do what you want. Same with the case - just get something that is well vented - heat is bad. Avoid those trendy Shuttle cases and clones thereof. They are too small to cool properly.
OS - Id suggest making sure you get a copy of a 64 bit system, be it linux, XP X64(discontinued) or 64 bit Vista. The performance increase is huge over 32 bit XP, even on identical hardware. With the right combination of hardware, and some patience, you can even put together a Hackintosh - An OSX Leopard running Mac clone, at less than half the price of a genuine Mac - there are articles on the interwebs explaining how to do this and the hardware required to pull it off.