Keep all your receipts!!! I don't know if you can still claim a tax deduction for long distance moves, but you'll definitely want to talk to a tax professional about it come tax time next year. I got laid off from my job in Florida and moved back to Indiana to find work and could have claimed a hefty tax deduction if I had itemized. That was eight or nine years ago, though. If you're moving for your job, you definitely want to look into this!
I've moved long distance three times (once from IN to WI, once from WI to FL, and then from FL to IN). I moved myself the first time, the company moved me the second (I think they used North American movers), and then I think it was North American movers the third time, as well. They damaged the paint on my car on the 2nd move, but they paid for all the repairs. They also damaged my TV and paid for that. Definitely look over EVERY big item they move for you and try all electrical appliances. If anything is damaged or doesn't work, file a claim with them right away.