Seconding the bit about education. I've been on both sides of this issue - before I realized I was trans, I'd had a couple of friends come out. I think part of the issue is that family especially tends to see it as a decision ("why have you decided to be a girl now, when you were always happy as a boy before?" etc), when actually it's a slow realization. The only decision is when and how you tell other people. But your family doesn't see the realization process, just the end result, the "I'm trans" announcement. Without knowing about the process, they don't get the benefit of all the education you've gotten along the way. Even people who are accepting of and knowledgeable about transgender issues may have trouble processing someone close to them coming out as trans.
I would suggest writing a letter or an email to give to your family members. In it, explain what it means to be trans - maybe talk about the current theories about what makes someone trans, and about how sex and gender are different - and explain terms like dysphoria. Talk about how you realized that you're trans, and how it feels to know that (does it make you feel more "right," does it explain things you did when you were a child, etc.), and how it feels to be invalidated by people you love. Ask them to explain how they feel. Avoid making accusations if you can - don't accuse them of hating you or not caring. They probably do care, a lot, in their own way. But if they're coming from a position of "being trans is bad/wrong/against my religion," then they may think that endorsing your identity will actually be bad for you. They need education, and they also need to understand how much they're hurting you, and how much you'd like a resolution. Having all that in a letter will allow them to go back and re-read it and maybe actually digest it, whereas just telling them isn't as effective because they may misremember what was said, or get so caught up in being angry that they can't hear you. Ultimately, remind them that being trans doesn't change who you are, because you've always been this way - you just have a word for it now.
When all else fails, though, you can't keep people in your life who are only going to be bad for you. You do need to set limits - for example, tell your family that you will only remain in contact with them if they use your chosen name and pronouns. You don't deserve abuse.
When people use the "God doesn't make mistakes" line at me, I usually tell them "I'm not a mistake, if God made me this way it must have been for a reason." Mind you, I'm not religious, but I do value the lessons about tolerance I've learned and been able to pass on because of being trans.
