I'm also in the process of teaching multiple friends how to play competitively. So far, it's not going too bad at all. So long as you don't get frustrated, it'll come eventually. Keep in mind though that my friends had already played the game casually for years. You may need to teach some really basic things, such as recovering, to your girlfriend before attempting this.
Note: As already somewhat mentioned, not a good idea to do this all in one day. The toughest part about teaching someone competitive Smash is probably motivation. You'll need to stay motivated to coach, and your girlfriend will need to stay motivated to learn. Make it interesting too; have her watch videos and play rather than read guides. (At least at this stage.) I don't recommend written guides until she's comfortable with most aspects of her character, but everyone's different.
First off, have her start by doing some doubles against even the easiest of CPUs, but with team attack on. This helped my friends learn to stay on the right target, and learn to not hit their teammate (me) at the same time. Kinda obvious here, but let her share stock, even if she SDs all 3 lives. It could also probably motivate her if she's winning against the CPUs all the time with you. Once your girlfriend feels confident against easy CPUs, adjust the difficulty accordingly.
Secondly, have her do some 1v1s against computers. Set the CPU difficulty equal to the skill level you believe she is on. I recommend replaying the matches. (Infinite replay hack could help, if you have it. If not, might want to set the stock time limit to 3 minutes.) After a few games, watch the replays with your girlfriend and start critiquing the matches. Make sure you aren't harsh at all though, it's not going to be motivating to hear that there are 10 things she needs to work on. Just point out obvious flaws, and then
explain to them how they could fix that in the future. Go back and keep playing more matches afterwards, it's one thing to get advice, and another to apply it.
Once your girlfriend has a good feel for the basics, I recommend going to training mode and practicing DI and Momentum Canceling. (Character specific moves chart for momentum cancelling:
http://www.ssbwiki.com/Momentum_canceling#Horizontal_momentum_canceling) Training mode is nice since you can set the percentage instantly to 100+%. After showing her examples of DI/Momentum Canceling, allow her to try it. Tell her exactly when you are going to attack. Once she has the general idea of how to do it, don't tell her when you're going to attack, so it's more of a reflex.
Now if I were you, I'd start doing more doubles matches, most likely against tougher CPUs. Ask her to try using DI and Momentum Canceling when she gets hit. Remind her in between matches to use DI and Momentum Canceling in indirect ways, such as saying "Nice job on the DI that match." or something along those lines. It should motivate her and reminds her to keep trying.
Here's where I'd start explaining some more in-depth things, such as ledge play, frames, priority, etc. But make sure you aren't overwhelming her with information. Teach the basics of one specific thing, and then play some matches against the CPUs. I don't know if this helps, but I picked up competitive smash by constantly beating up computers and practicing combos on a team with my brother, which eventually led to my interest in getting better. Play experience, at least from what I've noticed, seems to be the most beneficial.
Then, it's time for you to 1 vs. 1 her. Now, chances are it's still going to be entirely in your favor, but that's not the point. She will now be gaining experience against an actual player; something she may not have at this point. You don't necessarily have to let her win, but keep in mind that some people could eventually lose their determination if they lose every match. Or if your girlfriend has a competitive attitude, that could be exactly what you want. Some of my casual friends have a death wish for me on Brawl and just won't stop trying to beat me.
If you can, invite some other friends over, preferably around your girlfriend's skill level, so you guys can practice doubles against other players. Have her watch videos, mainly of her character at first, and then of others. Ask her to read match-up threads here on Smash Boards, and then attempt to apply it in-game. It is a bit different for doubles of course, but learning what the opponent is capable of is always a major plus.
Hopefully this helps, it's what I've done to help my friends get better, and it just might help your girlfriend too!