There probably is no "right" way, unfortunately. Everyone's situation is different, and you never really know how anyone will react.
In my experience, "blurting it out" worked well with aquaintances and strangers, such as when dealing with neighbors, nurses or dentists who I had to tell for whatever reason. With people who had no personal interest in me, it seemed best to just be matter-of-fact and not make any big deal of it. Those conversations never went past, "Oh, by the way, I'm changing my sex and going by the name Kate now."
But even with friends and relatives, I found that people really didn't care about the details or want explanations. They just want to know what to expect, that you're sure you're doing the right thing, and that you're happy. AND usually ask how they can help.
My one piece of advice: never make it seem like you're apologizing. Don't get into justifications and rationalizations, don't try to "prove" that your feelings are valid and OK. Just open your heart, and SHOW them how you feel. Give them the chance to truly know you at last, and trust them to cherish that gift.
~Kate~
