If you use movers when you move, you don't want to get ripped off. Follow Fox Around the House's advice to be sure you don't wind up with a moving company that takes your belongings or your money.
Before you've even asked for an estimate, ask if the company is insured. If they aren't, they may be cheaper but you have no recourse if they lose your stuff. Also, choose full-value insurance over released value. Released value only reimburses you based on weight and full-value covers the actual cost of an item.
Make sure you keep careful records of everything in each box. Assign a number to each box and then write down exactly what is in it. As the boxes are taken off the truck, mark them off your list and quickly look through the contents to make sure nothing is damaged. If a box is missing or something is broken, ask the movers for a receipt so you have proof and can contact the company for reimbursement.
Research the moving company's reputation by checking online review sites like Yelp and if you are making an interstate move, check out the US Department of Transportation's site, which any interstate moving company has to be registered with. There you can see their driving records, any complaints they've had and other information.
Finally, don't settle for an over-the-phone quote from your moving company. Be sure they come in person to give an actual quote for your living situation and belongings. You can also get a vibe for whether the mover seems professional.
When it's time for the estimate, go for a binding or "binding-not-to-exceed" estimate over a non-binding estimate. The company has to only charge you the amount on the two binding types of estimates, whereas with a non-binding estimate, they are giving you a ballpark figure and the actual amount could be much higher.