You ultimately need to decide what is best for you.
My advise is to take your time and make sure you have met all the people you can and will be working with and really have a feel for them, both superior and direct reports/support staff. Get a real feel for the culture.
Also, if you are moving, I am guessing a contract/loa of some kind is involved. Try to get a minimum payout equal to the remainder of a one-year salary, minimum 90 day parachute, or to have them reimburse you moving expenses to your next job if things don't work out within 12 months. If they are asking you to move for the job, it is fair for you to request some kind of commitment from them. Within a 12 month period is standard on most employment contracts I have seen. If you are a top level guy, VP of a division, Executive Director, or C Level, a two year agreement is more standard.
If it is a pay increase, and you take it, be sure to put the additional monies in the bank to safeguard yourself in the case things don't work out. Be prepared for at least 6 months of no salary. Then when the year is up, you have money in the bank to make interest off of, and new play money.
Don't be afraid to make the step, when you do though, be cautious and prepared. I have moved twice for a job. Once it was good, once it was bad.
Most importantly, don't buy an expensive showy car right after you start, at least don't drive it to work or tell co-workers about it. I have seen this happen countless times, and all it does is "xxxx" off your new co-workers, especially in a down economy where raises could be slow.
Good luck and congratulations.