When Autothrust is active, what methods or modes of speed control do you have? I noticed that the guys were making adjustments on the panel (such as heading, altitude, etc) for the autopilot. How does autothrust work in relation to this?
There are only two things we ever want from the thrust system. Either the thrust to maintain a desired speed, or a particular level of thrust (be it 'climb', 'idle', or a take off setting). The autothrust system works interactively with the autoflight system (which, in manual flight, is simply the flight director), to give an appropriate thrust setting/mode. So, if we select an 'open climb', the autothrust will select 'climb' power. If we select a particular vertical speed (V/S), the autothrust will give the power needed for a given airspeed.
The control being used on the mode panel is the speed dial. When pulled, it opens a window, and allows you to select a given IAS (target for the autothrust). When pushed, the window closes, and the aircraft goes back to 'managed' speed, in which case the speed target is generated by the FMC, and will have been pre-programmed.
Do you simply dial in a speed setpoint for approach and fine tune it as required, or is there other auto functions which control thrust (and therefore airspeed, altitude, etc.)?
Ideally you want the aircraft to be in managed speed. In practice, you're not in managed at all times, but the aim is to get back to it as soon as you can. In managed, during approach, the aircraft automatically slows with each new flap selection. When on final approach, managed will normally be Vref+5 knots. A function called 'ground speed mini' comes into play which automatically, and constantly, adjusts the target IAS to allow for the difference between the current actual wind that the aircraft is experiencing, and the tower reported wind on the ground...that provides some protection against windshear.