Yes you would just be doing class quests, but you can join your party member's class instance as long as you aren't the same class as him. It'd be a bit weird jumping around different places because different classes don't always get quests in the same areas, but it'd still be fun to duo, plus you'll get bonus XP (and social points....meh).
Edit: Oh you CAN actually both be the same class and join each other's instance. You need to both enable some option in the preferences menu. You would have to do each instance twice for you both to get credit, though. Probably better to stay different classes.
Edit: Oh you CAN actually both be the same class and join each other's instance. You need to both enable some option in the preferences menu. You would have to do each instance twice for you both to get credit, though. Probably better to stay different classes.