When my SO came out to me , I felt elated, giddy. I knew she was trusting me with this, and was thrilled she was taking steps to be the happiest person she could be. We talked a fair bit that first night. Accepting that this change was happening, and is going to be a part of my reality as her partner, that was the easy part.
The tough part comes mostly from me. In the days that came after, my head ran every scenario that it could come up with of things going wrong. By wrong, I mean my relationship going completely bonkers, either by splitting up, or my family finding out and taking my kids away. She was great during this period, gently reassuring me, and if I upset her (unintentionally), she has yet to mention it.
As far as advice for others learning to accept and support, I don't know if I have advice. I am very open-minded, and to me, her transition is just her becoming the beautiful butterfly she always knew she should be. I suppose my advice would be to remember that everyone has the right to live their own life, and our job as family/loved ones is to support that right as wholeheartedly as possible. If that isn't possible, I would recommend that loved ones seek understanding through counseling. And with those afraid for their family's stability and future, I would tell them not to be. Your future might involve transitioning, but it could just as easily involve a car wreck, a debilitating injury or illness....the future, and therefore stability, are not guaranteed.
As far as advice to those transitioning, it would be to remember that while you have been coming to your realization and discovery for years, your loved one has known for days, hours, etc. As happy as I was for my SO, I was overwhelmed by talk of hormones, and hair removal, and surgery, and rights and laws....everything. Even clothes and hair, it was too much. Between that and my head being insecure, I was downright miserable. With all of that, I was so happy that she was happy, and so afraid that if I told her to chill that it would be taken as me rejecting her. I ended up having a huge cry, and we talked everything out and have a better understanding, but that would be my tale of caution.
I hope my ramblings have helped some...and I wouldn't worry too much about reaching out here. I think it's great that you want to understand our side of it too