My family had a hard time taking the news when I came out in February this year. My dad said I looked sick and hasn't spoken to me since. The family friend who became my stand-in dad when my parents divorced only speaks to me to tell me "happy birthday" or "happy (insert holiday)", but that's it. He's deeply religious, and it's a double whammy because I'm a gay man too.
My mom took it pretty hard too. I don't think she's religious, either. We've never been a practicing family. She and I had lots of heated arguments in the beginning. I let the situation between us cool down for a while before approaching the topic again. She was worried it was a phase, or that I was making a mistake that would render my body disfigured for nothing. Her exposure to the trans* community is limited, so I was aware she didn't know very much about it. I decided that I needed to include her in the whole process for her to become comfortable with it and to become more aware. I listened to her concerns when she said she wanted me to speak to a professional therapist who had experience dealing with trans* identified people and to be sure it wasn't something else, like that I was a masculine woman. I kept the lines of communication open with her, and update her on anything as it changes. I tell my mom what I'm comfortable sharing with her about discussion topics with my gender therapist, and what my therapist has to say about things. I also encouraged her to help in deciding my name (my family kept joking that my name made me sound like a creeper lol). It took a while for her to understand, and she still has a ways to go. It's just this month that I was able to convince her to go to the SOFFA (Significant Others, Friends, Family, and Allies) monthly support group at my GLBT center. She lives in Texas and I live in Colorado thanks to my husband being military, so she went through the hassle of attending over Skype. My husband told me that she said she considers herself to have two sons and a daughter (my older sister), and finally started using my preferred name and pronouns throughout the discussion.
I think patience and compromise are probably the best bets in dealing with family. I'm not saying don't be yourself or don't pursue what makes you happy, but try to listen and understand their feelings on it too, and see if there's any sort of compromise that can be reached to make them feel less tense about the situation. Like my mom wanting me to get confirmation from a therapist. It was going to happen anyway, but me being open and honest with her about what was going on and my progress with the doctor definitely helped her realize it wasn't some phase, and that this wasn't going to go away. Best of luck to y'all, and I hope you find some way to open your family's eyes to what you're going through and that transition is right for you.