Basically the autopilot is a machine working on code. It's a simple device really.
As said above, pitch for flightpath and thrust for speed control. if you're low pitch up. it's far easier for a machine to understand so it's done that way.
As to modes. For Boeings, you will either be climbing level or descending. A LVL CHG command is the simplest, it will either give you climb thrust or idle and then pitch to maintain speed. VNAV climbs or descents are similar except the computer is being clever and switches from speed to mach and protects the envelope by changing speed at 10,000 or whatever is programmed into the FMC.
In V/S, again, it's simple. Pitch to give the correct VS and thrust to maintain speed as selected.
An Airbus is physically not much different. The programs and controls look different but achieve the same results.
Where you can get confused is the throttles- a Boeing will use the Autothrottle to move the thrust lever and the thrust lever tells the EEC to give you the required thrust.
An Airbus lever is effectively a thrust limiter. It will just use it know the limits you've selected. As far as I can figure out anyway.....