Yes, you can load a transport, move it up to 2 spaces, and unload it into enemy territory (occupied or unoccupied) during the Combat Movement phase of any round.
You can do this with or without any other naval vesels being a part of the landing fleet.
So as an example, Japan can land infantry in Midway just by sending a transport with 2 INF or 1 tank abord. Once the troops are landed, and your other combat moves are established, then those forces would fight each other until one side or the other was dead (there is no retreat from an amphibious landing). Or Japan could send a transport to New Zealand, drop troops there, and take it from UK without a fight.
Battleships can only fire their single bombardment shot though when a transport is offloading troops as part of the battle.
Hope that answers your question.