Tucson Water provides the Water Service in Tucson. There are a few other Public Water Providers in the Tucson area so you will want to double check and see who serves the area you are moving to. Oro Valley is served by Oro Valley Water for example. There are also lots of private wells scattered throughout the area serving individual homes or small groups of homes under "Well Share Agreements"
Tucson Water will usually open your account and switch the water billing over to you with just a phone call. Just let them know what day you want it switched over to your name and they can take care of it from there. The seller will also be calling them to tell them close their account. Be sure to call them or the will send someone out to shut off the water to the home.
You almost never need to be present for the water service to be switched over. There is no Water Shortage in Tucson but water conservation is always important in a Desert Community like Tucson.
Tucson Water is located in Downtown Tucson at 310 W. Alameda Street. Map
You should call Tucson Water a week before your scheduled closing date to make sure they have time to switch over your account.