I've told 14 of my friends so far. For each one I made a coffee or lunch date. I waited until toward the end so they could bail out if they wanted to (none did) but still have plenty of time to talk if they wanted. I opened it up by explaining that I wanted to share with them something about myself that I'd been keeping secret most of my life - I am transgendered. I would then go on to explain what that is, at least for me, and how I realize I finally need to be more open about it. I kept it positive and would let them change the subject when they wanted to. I explained to each that they are important to me and so it was important for me to tell them in person rather than have them find out indirectly. I think each of them appreciated that.
It helps if for a few days or weeks before you come out you can find a quiet place to visualize the meeting and the positive outcome. Each time negative thoughts come in, banish them and visualize the positive. Once you are comfortable with that, you can talk to your friends.
As for the doctor, I just told him I am transgendered, that after all these years I want to do something about it, and I'd like to start female hormones. No problem (at least with him).