Don't buy a branded computer, they _always_ priced much higher than if you build one yourself. The hardest part of building your own computer might be to choose the hardwares, and avoiding incompatibility, but incompatibility is rather rare. The best practice is ALWAYS choose the motherboard first, whenever building a computer.
CPU:
Well, the ones we're worrying most for CPU are usually games. Newer games nowadays need at least newer Pentium 4 or Athlon processor. Get a duo core processor if you wish.
RAM:
Web Designing don't need much RAM. HTML editing (WYSIWYG) needs around 64MB and Graphic Editing needs around 128MB to 512MB.
Surfing the net needs around 16MB-256MB, depending on your surfing habit. (if you're like me, who like to open around 60 pages open at the same time, you'll need more RAM)
Playing games... well, what kind of games? Newer games have really high RAM demand, and really high graphic card, but older games could run fine on a 32MB RAM. So, if you're playing those memory hogs games, you'll need around 1GB-2GB RAM (or sometimes 4GB).
Copy/Record Music and movies needs little RAM, around 64MB.
Flash, Maya, and 3DMax needs quite a bit, around 1GB.
So I'd recommend a 1GB or 2GB RAM should be enough.
Hard drive:
Seeing your use, you'd possibly need _at least_ 40GB. I'd recommend more, though, perhaps 80 or 120 GB
Graphic card:
Get a good ones. At least one with 256MB Graphic Memory.
Power Supplu (PSU):
Get ones with pure power. Don't buy a cheap PSU unless you want your system to be fried.
Audio Card:
Onboard audio card should be enough.
Monitor:
Since you're designing, I'd recommend something bigger than a usual monitor. Perhaps a minimal of 19" LCD monitor, use LCD if you're gonna work long hours in front of the PC so not to tire your eyes too soon.
Speaker:
Decent ones (you're not an audio freak aren't you? A decent stereo speaker shouldn't make much difference compared to a Surround Sound 7.1 audio arrangement).
Printer, Scanner, and other peripherals:
As needed.