Having moved across country 4 times with the major moving companies, I can honestly say they are pretty much alike. There is a lot of regulation in this area. You will have to have them come and give you estimates. These are actually fixed prices and the best advice I can give is to go with the people who you feel most comfortable with. I have never had a problem and the minor damage we had once was due to the company who stored our furniture for several weeks until the house was finished. They repaired all the damage to my satisfaction.
Another thing to consider is insurance. Is this a move from your employer? They usually will insure your goods. Your homeowners/renters insurance and car insurance may also provide some coverage so check with your agent also.
Relax, throw out everything you don't want to move...it is a pain to unpack all that extra stuff at the other end, and get organized. It helps a lot. Oh yeah, we have used Mayflower, United and American.