It sounds like you kind of do understand what was going on. The 'Auto' in autothrottle means it does it for you. Understanding the various modes is complex but in laymans terms you can maneuver with the sidestick and the engine thrust will adjust by itself. In a Boeing the thrust levers move up and down dynamically but on an Airbus they just stay in a detent - the levers don't move but the engine with adjust to keep your speed.
Big jets tend to keep themselves in trim so you don't need the rudders in flight. You generally only use them for maintaining the centreline on takeoff, during an engine failure, to sideslip a crosswind landing and on the rollout.