It was a while ago since I did ARR storyline, but I don't remember any cutscenes being in any of those dungeons that would force people to have to wait for you, not outside of the first time you enter a dungeon and it shows the area, and the final boss encounter cutscene, outside of that there's a couple instances towards the end of the story in where everyone is forced to watch all the cutscenes regardless anyway.

Though I guess I understand you're not knowing what's going on bit to some degree, though the 1-50 dungeons are very simple for the most part anyway, if somehow you're confused then just check what dungeon "quests" as they say what you need to do for anything, be it grab a key or some other item, or to activate what ever it may be. If you really want to run some of them solo and with AI, you can technically do that anyway by using the command missions which you will eventually obtain by ranking up your grand company rank to a certain point.