I go to Callen-Lorde and I started T there (I actually have an appointment this Wednesday for my shot and a doctor's appointment). I believe if you're 21 or younger you can get into the HOTT program (Health Outreach to Teens) which I got into last year when I was 21.
Basically when I first went there the process to start T was like this (to the best of my recollection): you have to fill out forms first, and then answer some questions with someone who works there. Then you get an appointment with one of the medical providers and they take your medical history and tell you the effects and risks of T, etc. I think they do a blood test at that point as well. Then you have one appointment with one of their therapists to go over the effects and risks again. The therapy appointment is pretty short and not a huge deal in my experience (and it's not really a therapy appointment, per se, just a meeting with a therapist). Finally you'll get another appointment with your medical provider and you will sign the consent form, get your prescription and have it filled in the pharmacy on the first floor, and then get your first injection by one of the nursing staff right after.
It's a pretty straightforward process and nothing to worry about. The only thing that can be tough is time between the initial appointments. They have a lot of patients and so it could take a few months to have your first shot just because they don't have any open appointments for a while (and the entire process to start T takes maybe 3 separate appointments). I should also note that I don't know whether they are accepting new patients or not (I assume they are and you can always call to find out if you haven't already), and because I started the process a good while ago it's possible changes have been made. Regardless, they will explain it all when you go. It's a great place, and everyone there that I've interacted with is really cool. It's probably the one doctor's office that I don't mind going to.
By the way, the HOTT program is on the second floor. You can go up the stairs next to the big reception desk you'll see when you walk in and it'll be right through the door at the top, straight ahead. Good luck, and feel free to ask me any other questions you have.