yep check the water.. since you will be on a spring you should have it tested anyways to make sure it is safe for you to drink, usually you contact your county. It will test for more then you standard water test equipment will. I've lived on wells much of my life. and moved a few times. maintaining your BB is crucial ime when moving tanks/fish. I usually put the fish in a bucket/bin and the BB in another bucket/bin. I wouldn't move the water in the tanks... I would just make sure the fish where all acclimated well, I prefer drip acclimation on moving.... also cut the fish off from eating for 48 hours prior to the move. And do a few extra water changes the first month and monitor your perameters carefully to avoid any water quality issues that may pop up. the old saying about an ounce of prevention....
The only exception to this is if your current water and the new water is going to be VERY different. ie if the particulalry water hardness. then it may be worth brining fresh water from your old tap when you move the tanks to do a 50% mix... Useing old tank water is likely to be more detrimential then helpful.
Even on a well I like to use a water conditioner that removes/neutralizes heavy metals, though we don't have chlorine/chloromines to worry about on a well we do have lots of smaller amounts of metals in our water. which testing your waters drinkability will test for these.