as a bisexual i've had this at another stand point, after all coming out is coming out no matter what it is about lol
here's what i can say, just do it, let them know, drop the ball, and then inform them.
Those willing to listen are those who really love you, those who turn away just because your a furry, well were they worth it? there's a giant chance they would have turned on you later...
THe point is your family, your real friends, will be ready to listen, and thats when you inform them.
People outside the furry circle see us as some kind of fetish, not a fandom.
Inform them that this is not really anymore a fetish than dressing up at a Star Trek/wars convention or the even more frequented Ren Fest
Yes there are people who get "off" to furry fandom, at the same time that same statement can be said about klingons, and im sure even the Hutts 0.0
Let them know that your just a hardcore fan of something!
and beleive me, it'd be better to be upfront about it and come out about it right away, even proudly, then hiding it like a dirty secret and having to "explain" it later when they find your suit/art or whatever
good luck!