Your city code may require someone licensed, especially for electric service to be turned on. It’s a personal comfort level for the other items.
Myself would probably hire an A/C tech to make sure the unit is operating properly after installation. You could probably ask the movers to hook it up but not turn it on and then have a licensed person come out after the electric is on and make sure it’s to code and freon levels correct, etc. Put in a proper pad for the outdoor unit, don’t let them put it on bricks or 2x4’s.
Water and sewer lines for me would depend on scope of work - if the lines are already run to the pad would probably let movers make the final connections assuming you have comfort with the materials they’re using (or you buy the materials). I had an uncle who bought a place and found out afterwards someone had used a garden hose to run water from the main trunk to an add-on bathroom. People do weird things when they’re not licensed.