I'm not in the USA so I don't know how you would go about it. I got my doctor to send me to a gender psych, who told my doctor to send me to the gender hospital. They have a six month waiting list and everything requires a second opinion, so after a year they told my doctor to prescribe hormones. By then I had been self medicating (definitely not recommended, but we do ...) for about three years. As soon as my doctor knew I was self medding - well before the hospital visit - she ordered blood tests. The important ones are liver function and PSA. Now I get liver function, PSA, estrogen and testosterone checked twice a year. I guess it would be more often if they thought there was a problem.
As for timing, after the one year delay to see the second hospital shrink it took their admin people 13 weeks to type the letter to my doctor. When the letter arrived my doctor phoned me and said the prescription was with their receptionist. I went straight there, took it to the pharmacy and got the hormones. YMMV!