I had my car shipped from Las Vegas, NV to Salem, VA by Dependable Auto Shippers in 2000. This page describes how DAS's service works and is probably applicable to many similar service providers.
I have included a link to only DAS above, because that's the service I used. There are many companies that offer similar services; I have made no effort to catalogue them.
When shipping a car, the key choices to be made are whether the company picks up your car or whether you drop it off. On the other end, you can either have it delivered to you or go pick it up at the shipper's terminal. My recollection is that it's about $100 to have it picked up or delivered.
In my case, I lived pretty close to the drop-off terminal and was able to drop off my rental truck close to a pick-up terminal, so I chose terminal-to-terminal service.
The auto carrier that was taking my car was right there at the drop-off terminal, so I stuck around to watch the loading process. It was the same kind of carrier that is more commonly seen delivering new cars, with room for about a dozen or so.
Particularly if your car is older, you really should consider selling it instead of shipping it. You may come out ahead on the deal that way. I was too emotionally attached to my car to consider doing this.
There are other services available for getting a car from one point to another. Some companies will put your car into its own trailer and take it; I'll probably use them when I need to ship my Ferrari(s) cross-country. Another option is called "drive-away," a paid driver drives your car to its destination. I didn't really look into that.
Before I started looking into this, I thought it would be a clever idea to load my car up with heavy/bulky stuff, such as books and clothes, so that I didn't have to move them myself. Every car shipper I checked, however, has explicit language in their contract that you cannot have personal belongings in the car.
Whatever choice you make, check with both the shipping company and your insurance company about whose insurance covers what. I seem to recall that my own insurance would have covered any damage to my car while DAS had it, but don't trust me on that point.