I have four indoor cats. I've moved twice with 3 of them, and once with all four.
As for how they react - they will stress out over the change in environment, but will adapt. Mine spent the first night in this house running around like absolute crazy. My uncle thought it was normal and wondered how I ever got any sleeping done. :lol:
Do little things to help them - put them in a closed off room while you get your furniture in (the furniture already smells right, so it calms them down), make sure food and water are ready ASAP, and don't forget the litter box. Don't forget to show them the litter box too!
Expect a few days of weird behavior - i.e., the normally affectionate cat to be reclusive, a silent bob becoming talkative, etc. They'll calm down soon.
This might make it sound horrible, but it's really not that bad. I'll be doing it again in about a year (when I get married, so I'll be moving them into a house with another person at the same time). They take a little longer than dogs to adjust, but probably will own the house long before you stop bumping toes going to the bathroom in the middle of the night.