My boyfriend and I also play together.
As we play played through the campaigns for the first time together, I always played the healer and he was the target-caller/director of our group in various capacities (he was an Assassin for our first time in Factions, Warrior in NF, Ele in Prophecies, Ranger in EotN...so he was always the damage-dealer). It seemed to work well for us, because he monitored the frontline and I monitored the backline for our group.
If neither of you are interested in being a Monk, I'd still suggest that you somehow split the frontline/backline aspect, perhaps one plays a caster class and the other a melee/physical attacker.
Of course, you should play the classes you are interested in and not feel forced to play any class!
As far as things to do together after the game is done...my boyfriend and I have done dungeons (Normal Mode and Hard Mode), worked on some titles together, completed elite areas like Urgoz/DoA/Underworld, saved up for special elite armors and weapons together, etc. There's 4 game storylines to play through though, so for now, that's a fairly large chunk of content for you guys to do currently. The high end stuff can be done much later, after you've both gotten plenty of elite skill captures and so on.
Good luck and have fun.
