That's a simple question, yet a complicated answer. When can it legally be turned on and off? Depends on the aircraft and company procedures. When do pilots turn it turn it on and off? Depends on the individual pilot and the situation.
The B757/767 autopilot can be turned on above 200ft after takeoff. Depending on the approach being flown it can stay on until the Capt wants to turn off the runway at the destination...assuming the landing was conducted from an ILS to an autoland. If the instrument approach was a non-precision approach, the autopilot must be disconnected no later than 50ft below MDA (fairly standard on most large jets).
The A300-600 is basically the same as above accept it can't be turned on until 1000ft after takeoff.
Most pilots like to hand fly the airplane after takeoff until 10,000-18,000ft then turn the autopilot on. Everyone's different. Most pilots also like to hand fly the approach and landing if conditions permit (wx, traffic, workload etc...)
In highly conjested traffic areas I'll turn the autopilot on as soon as possible after takeoff so I can divert my attention to looking outside for traffic. I think most folks will tell you the same thing.