There are some good answers here. I thought I'd add a few of my own.

"Do they use it to taxi and follow the perfect line?" No, for two reasons: One is that there isn't really a technology that exists that can lock on to an airplane and steer it around the taxiways to and from the gate... and if there was and it happened to malfunction, think about how dangerous that would be. Secondly, an AP isn't necessary for taxi. All you're doing when you taxi an airplane is steer it, brake it, and give it some gas...kind of like a car. With two pilots at the helm, they can do a much better job at this than a robot pilot.

"Takeoff is manual." Correct, in the sense that the pilots have to set the throttles, pull back on the control column or stick, lift the airplane off the ground, and raise the landing gear. At what point the takeoff ends and the climb phase begins is a matter of debate, but once the gear is up and the airplane has cleared any obstacles, this is really where climb begins and the AP CAN be immediately armed for flight control, though most airlines have a minimum altitude policy of at least a few thousand feet for safety, should the AP malfunction. However, modern airliner autopilots also have a feature called AUTOTHROTTLE, which is typically engaged as soon as the airplane begins its takeoff roll. It takes over throttle control so that the pilots can concentrate on other things like the control column, gear, and flaps. So there is a component of the AP that is typically active even before the airplane leaves the ground.

"When is auto pilot initiated after take off, how long after, is it used for the climb?" I think others have answered this question pretty well. I'll just add that even though some pilots might WANT to hand fly the airplane all the way up to altitude (believe me, if you're tired and have to fight a winter storm climbing out of O'Hare International, you will NOT), most airlines prefer and some require that you let the AP do the job. This is because the AP and all the associated computers onboard can "sense" everything that is happening instantly and more efficiently than a human pilot, and can (usually) fly the airplane more efficiently, which translates into better fuel economy, thus lower operating costs.

"Do they use it to keep in line with the runway for landing/ or how long up until landing do they use it till?" Again there were good answers here. It's mostly a matter of what the pilots want to do. The captain might let a young gun copilot take the plane from top of descent all the way to landing, a phase that might start 80 miles away from the airport. On the other hand, at the end of a 10 hour flight, with both pilots tired and shitty weather conditions at the destination, the AP will surely be used from top of descent to at least the final approach segment. (This typically begins when the airplane is pointed straight at the runway, has been slowed down a bit, and is "locked on" to the glide slope). From this point, the pilots may switch off the AP and hand fly, or allow the AP to fly the plane down to a hundred feet or so before going manual. What keeps the airplane ALIGNED WITH THE RUNWAY, however (assuming distance and/or weather prevent the pilot from seeing it), is part of the navigation system, not the AP. The AP just moves the airplane up, down, left, and right. It doesn't know where it's steering the plane, or how far it has to go. That's up to navigation and other computers, which in turn send commands to the AP. A particular runway will have an antenna array nearby that sends a focused signal to the airplane on final approach. The airplane's nav sensors receive this signal, so that the airplane can "home in" on the runway. As another person mentioned, this is called an ILS system. If the AP is in control, it tracks this signal to the runway automatically. If the pilots are landing the plane (happens most of the time), they too follow the signal via their navigation instruments, but make the flight control movements on their own.

With all us pilots weighing in, by now you ought to be an expert! Thanks all.