I know a lot of FTMs/MTFs who go to the Mazonni Center. What they told you is correct. They work from an "informed consent" model. You will meet with a licensed therapist who will do a "once over" kind of check to make sure you have the mental capacity to "consent" to treatment, then you will meet with an Endocrinologist the same day. They will give you a blood test. One month later you will come back and they will have you give yourself your first shot in front of them to make sure you can do it.
That is it!
It usually takes about a month to get in for your first appointment. So, you can expect to be on T within two months (or maybe a bit longer), from your scheduling of the first appointment. They take people with no insurance and their attached Walgreens will give you T cheap if you have no insurance (I think like $30).
If you are under 18 you will need signatures from both parents (or whoever has custody).