The OP guessed right; Imagine the thrust levers are transparent ghost levers which move up and down in the range from idle up to wherever you have set the actual levers. On the engine parameters display there are virtual thrust levers drawn on the N1 gauges or EPR gauges, and you can see these lines move according to auto-thrust commands.
The actual physical thrust levers limit the maximum thrust the auto-thrust can command.
In the first forward detent, the maximum thrust available is climb thrust, and this position is used for cruise and intermediate climbs when required.
In the second forward detent, the max thrust is Flex or MCT. Flex is the calculated max thrust needed to take off with the aircraft weight and meteorological conditions on the day, e.g. atmospheric pressure, wind and temperature. Flex is more than climb thrust but less than maximum available thrust, so it saves engine wear and fuel, (and noise levels).
MCT is maximum continuous thrust; normally used if one engine fails and you need maximum thrust on the good engine of a twin that can be sustained safely without a time limit.
The third forward detent is full engine thrust; maximum available thrust in today's conditions, and commonly referred to as TOGA - Take-Off and Go-Around thrust. TOGA thrust is time limited, usually to 10 minutes.
Other aircraft types use various external switches to select these various thrust modes, but Airbus FBW very cleverly and ingeniously combined the thrust mode select switches into the quadrant via a series of detent positions. So to select a particular thrust mode, you simply place the thrust levers into the appropriate detent. This is very intuitive. The levers do not move to indicate the thrust being commanded, but pilots do not need that, they will cross check the engine instruments anyway and will see the virtual thrust lever display telling them what is happening.
At any time, including pilot commanded manual thrust; pushing the levers forward will increase thrust, pulling them back will reduce thrust, so even though the levers don't move by themselves in auto, they still command thrust entirely normally and intuitively, (with a couple of caveats).
For a routine take-off; 50% thrust is set until all engines are stabilised at that thrust, then Flex/MCT is selected by placing the levers into that detent - "click, click". Once airborne and passing the thrust reduction altitude, the levers are pulled back to the CLB (climb) detent, where they stay for the rest of a (normal) flight, until flaring to land.
In some conditions and runway situations, TOGA thrust - "click, click, click" - is used for take-off.
With modern commercial airliners there is a device called a yaw damper which moves the rudder automatically to prevent a situation called Dutch roll, and the yaw damper also applies appropriate rudder movements in turns for turn coordination. So pilots do not need to move the rudder pedals - and shouldn't - unless they are steering along the runway; de-crabbing during a crosswind landing; or compensating for an engine failure during take-off.
The other thing you might have noticed is the Airbus fly-by-wire system. This would take a long post to fully explain, but essentially, the FBW looks at numerous inputs and feed-backs from the airframe as well as the pilot side-stick, and computes flight control surface positions to achieve what is being asked for, (while preventing certain exceedances). The FBW will automatically hold the last attitude commanded - within certain limits - when the side-stick is at neutral, and pilots need to be aware of this, as it needs a different technique to conventional aircraft to fly it properly. But it is not difficult to master.
The little square box on the flight director is the nose of your aircarft. Track and drift are illustrated with various other symbols.