I've had extremely good luck just being forthright and pretending I'm comfortable in the situation. It has gone pretty much the way Cynthia Michelle described it.
When I made my endo appointment, I called up the OB/GYN office where she works and when they answered, I just said (in my very male-sounding voice) that I'd like to make an appointment. There was a brief awkward pause, but then the receptionist recovered. She asked the standard, "Why are you coming in" question (i.e., it was not a "what the hell are you coming to us for" thing, just a question they ask every new patient) and I simply answered, "for transgender hormone therapy." She didn't give any indications that she was bothered or concerned with the reply, just continued setting up the appointment.
When I got to the endo, everyone was extremely polite. There was no one else in the waiting area, which it turns out is shared with a physical therapy office so nobody would have thought anything was up anyway, but it was still nice to have some alone time to get ready. The nurse called me in, took weight/height, and then went to a room to take vitals, medical history, etc. She asked why I was there, and I answered again, "I would like to start hormone replacement therapy," and we continued. Things were slightly but palpably awkward for the first 5 minutes or so, but as we talked and took the history, that went away. I think it helped that I was able to just have a conversation like I would with any other doctor, and I think my comfort rubbed off on her.
Then a resident came in for the first part of the "real" appointment. I was concerned at first because she was a young and extremely attractive woman, and I worried that might make it uncomfortable, but it was an unfounded worry. She was super comfortable and even enthusiastic in a way that really put me at ease. She again asked (3 times so far if you're counting) what I was here for today, and I gave the usual answer without fanfare. I don't recall what she replied, but it was positive and non-judgmental, and she then proceeded to do a 10-15 minute interview about various elements of my medical history, gender identity history, and current situation.
Then the real endo came in, and we just talked nuts and bolts (excuse the imagery).
At no point did anyone judge me for what I said, push back on the truth of my statements, etc. They were at worst neutral, and at best super-positive about everything. The same was true at my follow-up 3 months later.
