I agree with USAFRet. If you need or want the portability, get a laptop. If your computer wont be moving, buy a desktop. I wouldnt recommend building one based on what you said in the OP. It sounds like you just want to have something right now to play games.
Building gets you more bang for your buck, but it can take time and cause frustration if things don't work perfectly. Sometimes one or more components either don't play well together or they may be defective right from the start. It's not that common, but it does happen. Building a PC means you'll have to do some research if you don't already know exactly what you want, and you'll have to wait for each component to ship to you if you don't buy local.
If you want a PC right now, buy one. You'll pay more for less, but you'll have less headache because it's ready to go out of the box. If you want to have a fun experience crafting your rig from the ground up, build one and you'll be happier in the long run.