Many years ago when people were allowed to view the flightdeck in the air, people would expect us to be fighting the controls like James Bond to keep the plane in a straight line. People now know we have autopilots and with that came this myth that pilots don't do anything anymore, they just push a button and the plane starts up, taxies out, takes off and lands on it's own.
Aircrat today are very complex machines but they are not capable of taking off by themselves and getting you from A to B. Pilots always do the take off and will fly the initial departure until they have had enough and put the autopilot on, we then control the aircraft through the autopilot during the cruise because it allows us to concentrate on other things and frees up our capacity. Aircraft do have the ability to land themselves but this is only carried out if the visibilty on the ground is zero and we cannot see a thing because the human eye cannot focus on the instruments and then in the very last second be able to adjust to the limited view outside due to fog etc, this hinders the pilot ability to find his bearings it is to dangerous.
97% of approaches and landing are carried out by the pilot. The autopilot is limited to strong winds were it simply would not cope with all the control imputs, the pilots limits in strong wind are much greater because we can judge things better than a computer and we can also react more quickly should something non normal occur.
So the answer to your question is yes planes can land themselves but only if the winds are not strong and is generally only used when the visibility is virtually zero, 9 out of every 10 landing will be carried out by the pilot.
Sure come fly with us and if you are out of Bristol and hear me on board don't be shy ask to come up and see the flightdeck, easyJet pilots and cabin crew are looked upon as some of the best in the industry and will always be more than happy to answer any questions you have.