I moved over the summer. I moved about 350 miles. I moved a one bedroom apartment. I paid around $2,500. This was the honest estimate that I got from about 3 reputable movers.
One thing I will say. Get what's called a binding estimate. A binding estimate holds the movers to the price they quote you. There are many, many horror stories out there about moving companies who get your stuff to your destination, only to add on some extra costs. You don't get your things until you pay them, and it's perfectly legal.
Check
this site out before you contact a mover. We found a reputable mover and learned a lot about our move before we made it.
Good luck!