This is something I'm still not clear on. After a plane takes off, do the pilots pretty much engage the autopilot right away and then fly it using the AP controls? And when they land, do they let the autopilot bring the plane down to a few hundred feet and then land it manually?

Depending on pilots, airline procedures, aircraft, etc.Some 767 pilots for example like to always hand-fly the aircraft below 10,000 ft, some even hand-fly all the way to cruise.There are different categories of instrument landing systems.In CATIII a properly equipped airplane on a properly equipped runway can land all on its own - used very rarely however.Most typically pilots take over controls at around 200-1000 ft and land manually.Some pilots just love to do as much hand flying as possible - they would typically do it during takeoffs and landings, climbs, descents since cruise is the one segment of flight where pretty much everyone uses AP.Michael J.

I'll echo what Michael J. said, but for military pilots. Most of us, (C-130 guys that is) will hand fly up to cruise altitude, around FL180-200, then once the plane is leveled out and TRIMMED, trimming is very important for our antiquated AP, we'll put the autopilot on and cruise, until the descent. Most guys like to handfly the whole descent, some use the autopilot till around 10,000 or so. All takeoffs and approaches are always hand-flown. I'm not even sure if you can fly an autopilot-coupled approach in the -130, lol. Lobaeux