There are going to many problems but it will be worth it in the end. My fathers been dead almost 25 years and I found out a short time ago that he never accepted me becoming a woman. As for my mother, I talked to her on the phone all the time and she took years to get used to it. Still both supported me and were civil. Also get ready for some more sleepless nights because any time emotional events happens, you may pay for it that night.
On passing in public, the best thing you can do is not let it bother you. If you show nerves, it will confirm what they suspect. Just think you are a beautiful woman and be proud of it.