Crate training is definitely a must for several reasons. One is, that it's the safest way for her to travel. The other is, that she needs to be able to feel safe and relaxed in a crate, should she ever need to be left at the vet's over night. However, it is not necessary for house breaking. I know it is commonly used in the US, but it's not used at all where I'm from. Both of my dogs are perfectly house broken without crating.
My method for house breaking would be this: Decide a schedule for potty breaks. To start with, I'd taker her out often, so as to avoid accidents. Perhaps once every two, or every three hours would be good to start with. When she pees or poops, praise and reward her with a treat. This way she learns to associate going potty outside with getting a reward. If she has an accident indoors, never punish or scold. Just quietly clean up the mess, using an appropriate scent removing product. Any remaining scent will encourage her to go again in the same spot. It's instictual. The accident is a sign that she needs to be taken out more often. A rule of thumb is that puppies need to go after napping, playin and eating. So build your schedule around those things, and like I said, take her out often. The road to success with any dog training, in my experience, is to make it really easy for your dog to do the right thing, and difficult to make a mistake. Since she doesn't yet understand that she's supposed to hold it in, you shouldn't put her in a position where she must do that. Much better to get her used to going outside, and then to wait longer between walks. All dogs, even adults, ought to be taken outside at least every 6 hours during the day. Puppies, a lot more often.
Apart from that, I agree with what's been said about training. Positive, reward based training is the way to go. There are great books to read. I think Karen Pryor's books would be good. At this stage in your relationship, the most important thing is to create a bond of love and trust. Lots of cuddling and affection is good. Playing is incredibly important. When your dog learns that everything good comes from you - affection, praise, treats, food and fun - and that you can be trusted, training is easy. I think people put way too much emphasis on teaching commands, and way too little on building the relationship with the dog. If the relationship is there, you'll get a dog who wants to be with you (which makes off leash walking possible) and who wants to please you (the foundation for all successful training).
I wouldn't worry too much about your pug trying to get in the bathroom and following your girlfriend around. That's how pugs are. Both of mine follow me everywhere. If I let them, they'll be in the bathroom with me. If I take a shower, I'll find them napping on the bathmat, wating for me. This behaviour is very typical for the breed and doesn't necessarily mean that the dog will have separation anxiety. But if your pug has never been taught to be alone, that should be a priority for you once she has settled in a bit more.
My preferred method for training how to be alone is this: Begin with closing off one room with baby gates or similar, preferably the living room or some other room where you spend a lot of time. Put the dog in that room and give her a chew toy or similar, to keep her distracted. Without making a big deal out of it, leave the room for a minute and then come back. Repeat this so many times, that the dog gets used to you coming and going. The fact that you always come back will teach her that you do always come back, so there's no need to worry. Once you're able to leave the room for 10 minutes or so without her kicking up a fuss, start leaving the apartment. Again, just for a minute. Repeat multiple times. You coming and going should be undramatic. There's no need for a big greeting ceremony when you return.
Proud mama to puggies Winston and Ozzie, Slatan the Cat and Zino the horse.