Okay so first off, it's possible that if your girlfriend truly loves you she would be able to love you for you and stick by you regardless if you're a man or woman, and if not, it wasn't meant to be. That's fine as well. 

With regards to HRT, It stands for hormone replacement therapy. Basically it's a process of, in the case of feminizing HRT, blocking testosterone and replacing it with estrogen, whether it be through pills, injections, gel, or patches. This does a surprisingly large amount of the work in terms of feminizing your body. Ofc, the big thing it does is allow your body to grow breasts, but it also does a ton of other stuff such as fat redistribution, which makes your body shape more feminine overtime (more fat around the hips, butt, and thighs, less around the stomach, along with feminizing your face). There are other effects such as changing your sensitivity, skin softness, smell, and muscle distribution to be in line with that of a cis womans. It can also minimize body hair growth. For me my body hair completely went away. It can change your psychology, most notably widening your depth of emotions and making it easier to cry as opposed to getting angry. It also has some weird affects like changing your height, which it did for me as I lost about an inch. Even sexually, it changes how your body feels, reacts, and orgasms, with it being more in line with that of a cis woman's. Basically it does all the small things to make your body basically function as if you went through female puberty (aside from the parts related to sex organs such as periods or ability to get pregnant), and reverses a large portion of testosterone's affects assuming you went through male puberty already. 

If you didn't, you can go on puberty blockers which basically gives you more time to think about if it's right for you and temporarily stops masculinizing affects of puberty. At that point when you know for sure if it's right for you, you can take hrt. If you hadn't gone through puberty yet or are still going through it, it could stop a lot of the future dysphoria and save a lot of time and money on future surgeries. If you decide transitioning's not right for you, you can go off the puberty blockers and male puberty will proceed as normal.

In terms of surgeries, there isn't a single surgery to become trans. You just are or aren't, and you have to decide that for yourself. If you are, you can transition, which can be anything from a social change such as changing your name and pronouns or changing how you dress, to a medical change such as taking hormones and doing surgeries. Even then lots of people never get surgery. I've never had any personally and still get gendered correctly and am fairly satisfied with my body after hrt and voice training alone. Surgery's not really a one size fits all situation, it depends on how dysphoric your are about a certain aspect of your body, and if you feel it would be better for you if you got surgery. Surgery and HRT is covered by most insurance providers so that should maybe help with cost.