It's difficult, probably the most difficult thing you'll ever have to do. No matter what you do you're going to be at least somewhat anxious, however there are several things you can do to help reduce that anxiety to a more manageable level. First you can test the waters a little, try and see how they feel and how knowledgeable they are about trans stuff. If they're more secular and liberal, then generally you can expect them to be more able and willing to understand.
When you do tell them, make sure you sound confident in your identity and what you intend to do. If you have doubts then try not to show them, since they'll use it as an excuse to dissuade you from doing anything. It may also be helpful to bring some printed information explaining what being transgender is. Mentioning things that they may have noticed about you and giving it as evidence as being trans can help as well.
I don't have a huge amount of relevant experience at coming out since I already knew how my mum would react before I told her. My brother is trans (ftm) and he came out a number of years before me, so I was able to tell my mum without much to worry about. I still felt really anxious when doing so, however! I hope that when you do decide to tell them that it goes well for you, I'm sure that others here can give some more helpful advice for getting over your fears. Best of luck!