Flight instructor here.

I've seen autopilot systems that have basically every level of complexity you can imagine. A lot of Cessna 172s were equipped with a single axis autopilot that can only control the ailerons and can only maintain wings level. Others have control of the elevators and can do things like altitude hold, or ascend/descend at a given rate. More modern ones have control of all three axes and integration with the attitude instruments, and can do things like climb to an altitude and level off, turn to a heading and stop, or even something like fly a holding pattern over a fix. They still often don't have any control over the power plant, and small aircraft typically cannot land themselves, but there are autopilots installed in piston singles that can fly an approach to minimums.

And that's what's available on piston singles; airline pilots seldom fly the aircraft by hand anymore.