My transition started with a couple's session with a psychologist, because I thought it is important for my wife to be there to realize the seriousness of gender dysphoria and what kind of treatment would be needed. After the first session, my wife understands that I'll be in the care of many professional medical staffs, i.e. psychologist, psychiatrist, endocrinologist... etc. She also realized that transitioning doesn't necessary mean that you have to go through from HRT to top surgery to bottom surgery, those are "treatment options" which "patients" can choose. She felt a lot safer about the whole situation. Then he referred me to a psychiatrist for proper diagnose, and I only went in by myself because there's really nothing my wife would change the outcome of the appointment. He then referred me to an endocrinologist for T and a surgeon for top surgery.
My situation might be different than most of you guys because I'm in Alberta, Canada. My healthcare system actually covers the appointments with the psychiatrist and the endocrinologist, as well as top and bottom surgeries with specific surgeons as long as you are willing to wait.
Anyway, I believe it's a good idea to have your girlfriend go with you to your first session, perhaps until she feels more at ease about the situation. Sometimes information coming from a third party makes more sense than just you says she says.