I would say that WoW is absolutely worth getting into now. The game has never been more newbie/casual friendly, and while leveling up to 110 SOUNDS like a daunting task, it only really feels that way the first time you do it.
However, I have some suggestions that might improve your initial experience.
1. As far as which class to choose, I wouldn't so much say which one, but which ROLE. Healing and tanking might be better served after you've seasoned yourself a bit with the mechanics. DPS (damage per second) would probably be a much more exciting (and easier to learn) idea. Fortunately, EVERY class is capable of being a DPS. Really, I would just go with whichever one has the most appealing lore to you (although, I would avoid picking either a Demon Hunter or a Death Knight to start with, both classes are considered "Hero" classes, and are intended for more advanced players).
If I HAD to suggest a class, the Hunter is probably the easiest class to learn, and the easiest to solo content with.
2. Join a social guild as soon as you're able. Having people to talk to and bond with are essential to getting the most out of WoW. Plus, having people nearby who can answer your questions is always a plus. If your girlfriend's guild is sociable enough, that step is already checked for you.
3.Take your time. There's no rush to get to max level. Even as the current expansion's endgame content goes on, there are always catch up mechanics to make sure that even if you join late, you'll be able to (sooner or later) catch up to the other max level players in whatever you decide to do, either raiding or PVP.
Moreover, while you CAN spam dungeons at lower levels to level up faster than questing, I wouldn't really recommend doing this your first time around. Do some on occasion, sure, but do some normal questing, explore new zones as you level up, maybe even some low level PVP battlegrounds. Just try to keep shaking up your experience. There's alot to do in Warcraft, no need to burn yourself out sprinting to endgame.
4. Save your level 100 boost. If you do end up getting the Legion expansion before you reach level 100, you'll be given a level 100 boost to use on any character. I'd say, save this boost. Boosting up to the new stuff sounds like a good idea, but really, it'll put you in control of a near-max level character, with no idea how to use them. Spending time leveling up the old-fashioned way will help you learn how to play your class, and learn the game's mechanics without as much pressure to perform that comes from doing current content. Plus, you can use the boost on another character later, if you decide to reroll to a new class later on.
5. Don't level up with someone else. I know you may be tempted to partner up with someone (probably your girlfriend), and reach max level together, but personally, I'd refer back to point 3. Go at your own pace, whether that be faster or slower than someone else. Again, there's no reason to rush, and conversely, you shouldn't feel pressured to wait for someone else to catch up if you're going faster than them.
Just my tips on getting the most out of your first character.