I've shipped all kinds of stuff, such as books, clothes, computers, and televisions, by almost every major shipping carrier out there. This page relates some specific tips for when you're shipping your belongings.
The links above have a lot of good information about the various shipping options that are available.
There are other shipping companies other than USPS, UPS, and FedEx. I'm not trying to slight them by not linking to them. This isn't an effort to list all the available companies.
Every shipping company offers a variety of shipping speeds. For a premium, they'll deliver almost anything as fast as you like. When you're moving, you usually have plenty of time to let belongings get there. The slowest and cheapest shipping option is usually the best when you're moving.
Books and computer media can quickly add up to heavy packages. The US Postal Service offers a special option called "Media Mail" (it used to be called "Book Rate") that will be much less expensive than other shipping methods. You can only send books and other media this way.
Shipping companies don't handle packages as gingerly and carefully as one would like to imagine. Package breakables and electronic equipment accordingly. If you have the original packaging for your electronic equipment, by all means use it. (This suggests that if you think you'll be moving in the near future, you may want to keep packaging.)
I usually hand-carry my computers. If I had to ship them, I'd probably take the hard drives out and hand-carry those. In any event, make sure your data is backed up. In today's security-conscious environment, there is no telling how strong a magnetic field to which your packages may be exposed.