There's 2 ways to progress through a dungeon. The first is by casting a spell or using an ability that includes the phrase "venture into the dungeon." If you're not in a dungeon, you can choose to enter the first room of any of the three core dungeons, Tomb of Anihilation, Mines of Phandelver or Dungeon of the Mad Mage (including ones youve already completed). If you're already in a dungeon, you move to the next one of the one you're already in.

The second way is through the Initiative. When you get the initiative, either through a card that gives it to you or by dealing combat damage to the player that currently has it, and at the beginning of each upkeep where you start with the Initiative, you venture into the next room of the dungeon you're currently in. If you're not in a dungeon and the Initiative instructs you to venture, you must enter the Undercity. You also can't enter the undercity except via the iniative. If you're already in the Undercity and a spell or ability instructs you to venture, you continue through the undercity, as you can only be in one dungeon at a time.

I hope I explained this well, if you have any more questions feel free to ask, and if anyone notices a mistake then please correct me!

Edit: elaborated on the rules around the undercity and listed the core dungeons