Speaking for myself in our Citation, the auto pilot goes on after about 500ft on takeoff, and depending on clearance will usually be set to fly a certain heading and maintain a certain vertical speed. The initial cruising altitude is set (usually as assigned by ATC) and the AP will capture that altitude. Enroute, the AP will be set to Nav mode and will follow the route programmed into the GPS as assigned by ATC. Altitude changes are usually done through setting the new altitude, and making a power change as necessary. The auto pilot will also fly much of the approach, and I'll fly the last 1000ft by hand. The Citation actually hand flies very nicely, and I have flown an entire flight this way. This is typical of GA operations in smaller airplanes equipped with auto pilots.
Some Airlines have autopilots that are more advanced and can fly 'Category III' approaches, allowing the autopilots to land the plane without pilot intervention. Ths requires special equipment and pilot certification.