Hmmm.. I have had many a friend in your boat, and also have been on the receiving end of this kind of thing.
For my friends, most of them were best able to have their family understand it by explaining it calmly, and with pride. I think if you let them know that you are not saying it for some sort of trend, and it's not some tragic thing they caused (cause they will try to blame themselves), and you carry yourself in a responsible and sensible manner, it will be alright.
Also, as a recipient of such news, I have a small story, with a point following it.
I once fell deeply in love with a boy, and he lived in another country. We met via Roleplay, on Gaiaonline. I was simply his friend for nearly a year of knowing him, but eventually we both confessed deep feelings and began a relationship. Now, you may think what you want of online relationships, as I myself critique them a lot. However, I was with him for nearly two years, until I broke it off with him due to hurtful circumstances. But, after a messy period of being unable to talk to each other cordially, we became fast friends once more, and even now are good friends.
Getting more to the point, recently after a period of not being able to talk to each other, he came to me and told me that he (for a long time) had reached a point where he found himself to be a female trapped in a male's body. But not just a girl, but what is commonly known as a dickgirl. A transsexual, of sorts. He had a new name he secretly is going by, and has not told any of his less trusted friends or a single family member. And me, having only known him to be a great Guy, and one of the most chivalrous men I have ever met, was both confused and a little hurt when first finding out. I have no doubt it was mainly because I had loved, and still do love him/her. It also, though, equally hurts to know how he pictures himself currently. He sees his male body as ugly, and an abomination... whereas I have always loved it, and he, and was sitting there listening to him crush everything he used to believe in himself to sad smithereens.
But, that's enough of my personal life, truly. The point to that little insight is that... The hardest thing, for me, was not accepting and understanding his new found gender and sexuality... The hardest thing for me to do, was learn to separate the past, and all I had cherished with him, and all I had thought of us together... separate it from her, a girl talking to me from a male's body. It was hardest for me to separate those two different people, and then put them back together to create not a person with a gender, or sexual preferences, but a person who I loved, and who loved me. Questions that went through my head were not 'why..? that's disgusting!' nor were they 'holy cow, that's unnatural!' But instead, they were more like 'does that mean my memories are all lies..?'
And so, your family may not respond completely well, but it may not always be for those judgmental reasons, even if they lash those out at first. All of that supposed disgust may simply be their defense as they sit there asking themselves 'did I really know my daughter/sister/etc...?' and 'are all of my memories with her lies..?' because they cannot yet comprehend that a mental image they have of you, is being tampered with. It's harder than it seams, for a lot of people. Again, for me, I did not care that he was claiming to be a woman. I cared that that may then damage all I had ever had with my dear friend.
So just.. be gentle? And remind them that they are loved by you.
Well, that's quite enough out of me... Heh, definitely. Thank you for reading.
Forsaken