I would suggest using United Van Lines. I'm a little biased, though, because I have been a subcontractor for them for 11 years now. Mayflower Transit is owned by United, so things should be about the same as far as cost. Just make sure your estimator actually takes the time to do a thurough estimate. If it seems that he's in and out in a matter of minutes, that's not a good sign. A binding estimate is a good idea. And make sure you get your estimator's cell phone number, as well as the phone number of his home office.
To make the move as stress free as possible, be prepared before the packers arrive. Make sure you have everything you need to use for the next few days separated so they won't get packed. Nothing makes the packing day go worse than if the customer isn't ready. It takes much longer than it should, and adds stress to everyone's day.
If you are doing the packing yourself, make sure you are all finished with the packing before the movers get there. Make sure all the boxes are taped shut, and are packed as full as practical. And oh yeah, don't pack heavy things in large boxes! Make sure to have a few extra boxes on hand on moving day just incase something was forgotten. (It usually happens).
If the estimator offers the "unpacking service" at the other end, this usually means the unpackers simply empty the boxes onto any flat surface and haul away the boxes and paper. It makes one heck of a large mess that way. If you unpack yourself, I suggest unpacking a couple of your largest boxes first, and use them as paper receptacles for the packing paper. Then break down each box after it's empty. It'll save you lots of space.
Let me know if you need any more advice. Good luck with your move!