I think the confusion with transports comes from non combat and combat moves.
A transport normally only moves during non combat, it can only move during the combat move phase if it is participating in an amphibious assault.
The restrictions below, on transports apply only when they are moved during the combat move phase if I understand correctly.
Simply put: if you move a piece during the combat move phase, it must participate in a combat that is resolved during the conduct combat phase.
That means you can not pick up more then you intend to unload in that combat move turn.
However, if you start the turn with units on a transport (loaded previous turns not this turn) then you can still use that transport in your combat move phase, and any or all units not loaded this turn can still land with any units picked up this turn(if any) to support the amphibious assault.
If you had one unit on board (like an aagun that does not move during combat move phase), you can still use that transport during combat move phase to pick up an infantry and drop that infantry during an amphibious assault while leaving the previously loaded unit on board.
You can only have one unit loaded prior to the current turn remain on board a combat moving transport(if there were 2 units already on board not participating, this full transport would wait until non combat to move). If a previously loaded transport moves during combat move phase, any unit picked up on the way during that turn must participate in the battle. Unless you are forced to retreat, thus aborting the combat and leaving all units stuck on board until the next turn, as they may not be unloaded during non combat phase as the transport is considered done with its move prior to non combat and thus the transport is unable to unload until its next turn.
Edit: A transport moving during combat move phase may not pick up units that can only move during non combat(such as an aagun) as it is the combat move phase. These units can stay on board if they had been previously loaded and the transport is used to pick up a combating unit.
Normal non combat moves: You may pick up from 2 or fewer locations up to 2 units total as long as one is an infantry and then you may drop off one or more units to only one location and if you choose, may leave a unit or two on board.
I hope that wordy description helps and is right.
Edit: strike-through above per Krieg below.