Moving day is hectic enough. Having moved several times with animals, this is what I do. On moving day, I confine the animals to one room, with their toys, litter pan, water food etc.. The night before, I pick up everything that I know is familiar to the cats, their food bowls, toys, blankets, etc..I put all of this into one box and add canned food, a manual can opener, plastic spoons, paper towels, cat litter, everything the cat is going to need in the new home, that I might not have time to go looking through all the boxes to find. I set the box aside to travel with us in our vehicle, not to be put in a moving van, and I mark the room where the animals are clearly off limits so they don't get out. I leave a radio playing for them as well.
Once we get to the new place, I keep them inside however long it takes for them to be acclimated. There is no set time schedule, where you can just mark your calendar and say, okay, 2 weeks is up, time for freedom! I keep them in one room for about 2 weeks and if possible instead of putting a closed door betweeen them and "us" I get a screen door and put that up instead. That allows for them to hear the house better, get used to the new smells etc. Also before I let them out in the other part of the house, I take my handy dandy blacklight flashlight and shine it all around in the middle of the night looking for old pet stains (which I nearly always find) Then I get to work eliminating the old stains before my animals find it first.
About the new introduction, I would wait until your cats are comfortable in their new home before pressing on them yet another strange thing to get used to.