Is there another way to listen to quest dialogues or watch in-game cinematics if the party leader constantly skips them?

I believe anyone in a multiplayer game can skip dialog or cutscenes, and the skip effects everyone in the game. Sometimes it's just enough to say "hey, I haven't played this yet, do you mind?" - but at other times you might want to just go solo for your first playthrough.

If, however, you're committed to playing with others, and those others are being unreasonable-poopy-head-dialog-skippers, you can review the quest dialog and cinematics at your own pace whenever you wish.

In addition to using the main menu cinematics option (as has been already suggested) to see the in-game cinematics that occur between chapters and at major plot milestons, there is a good database of all the dialog in the game, for each class, on the Diablo 3 DB pages.

For example, this link will show you the quest dialog for the very first quest in the game.

I tend to play Diablo 3 via a "non-Steam game" shortcut, which gives me the Steam overlay. With the overlay's browser, you can be on the DB while you game.

In-Game cinematics can be re-watched from the main menu. See this question: Where can I view cinematics I've unlocked?

Quest text/dialogue however cannot be.

To prevent issues of people joining your game you can always turn off the auto-join feature in the Social options of your game (Options -> Social -> Checkbox that reads "Allow Quick Join"). This will prevent people from joining your game and making you miss the content.