I got a staggered start, I was on Spiro for three months and Estrogen for one before going full-time. I had an RLE that started just about a year ago now that went through my last semester of grad school.
Getting used to being seen as a woman, as well at the rights and restrictions that it affords, is a real good idea. I found learning I could function doing day to day things such as going grocery shopping or to meetings with Teacher's Assistants really encouraging and it set me starting to believe I could have a successful transition.
As for getting over my fears, I had to get out and face them. I didn't have a car when I went through my RLE, and even before my RLE started when I went to see my therapist, it was either a walk through Downtown or a bus ride for me. I remember I would look at women wearing short skirts on cold days and thinking, "WHY!?!?!" Then I went, wearing one with leggings of course, for one of my walks to a therapy session. I learned why very fast. It's one of my fondest memories.
I don't know that I recommend "dive right in" route however. It was stressful because I had no safe place to go when I got clocked, so I had to power through it and suck up my confidence so I could get home. I didn't even have friends or anyone really by my side, physically at least, as when I went through all that as my SO was 300+ miles away.
One of my early times out to a general public space was to a movie. Went and saw the Tron sequel. I was terrified over having to talk to the ticket person, but I got through it and had a nice chunk of time afterward to be out in the world, as me, but in the dark. I did other things before that but that one was the, I wanna say least stressful even with the having to talk part. Oh shoot wait! Actually the second thing I did on my very first outing was go see the first rebuild of Evangelion Movie in Chicago. So yeah, I do recommend going to see a movie.
Congratulations on starting HRT! And good luck!