How do I share quests with my friends in Star Wars: The Old Republic?

If you are different classes, you can help each other with quests but the class-specific quests cannot be acquired by any other class. If you're the same class it's similar, but you can only help each other with those quests if you've turn on the option Preferences > Social > General > Allow Access to Same Class Story Areas. However, the quest will only count for the character who entered the instance first, and after you both leave you'll need to enter with someone else first and repeat it again until you've all done it.

In either case, you should to be able to see each other's cutscenes if you are present in the same instance. If it's a scene out in the world (or on their ship, I think) you won't be able to.