I can help you on the topic of what to expect. The waiting room will be like visiting any other doctor. They'll want to see your license and insurance card, etc. No one there will be staring at you or silently judging you for being there. The receptionist sees nothing but people with psyche appointments all day long, and the other patients are there for the same reason. They'll probably have you fill out paperwork about why you're there and your general medical history.
The doctor will want to do introductions with you, and then they'll probably ask you to explain in brief why you're there. (Even though you just wrote that down. It's a thing.) If they're a good psychiatrist, they'll keep their questions focused on the areas you've identified as wanting help with. They should not be asking nosy questions about your transition, unless you've brought it up as something you want to talk about. Gender identity is no longer considered a disorder, and they should definitely not try to "cure" you of being trans. (If they do, find another doctor ASAP.) Toward the end of the interview, the doctor will probably ask you if you have any questions. You might want to write some down so you can remember them when they ask. Some questions that might be worth asking include: "Why did you choose to try me on this particular medication?" (They will probably start you on something.) "How does the medication work? If I have a bad reaction to the meds, or if I just feel worse in general, how can I contact you during non-business hours? How long should I expect to wait before these meds start to work? If they don't help me, what then? What is the usual prognosis for patients like me?"
I also like to check out a new doctor's office, looking at any framed diplomas about where they went to school, what board certifications they have, what books are in the bookshelf (indicates areas of specialty), office decor that clues you in as to their personality, etc. Some psychiatrists share offices, though, so the place may be pretty generic.
A good shrink is like a miracle worker. Here's hoping you get one of the best!