Female and bisexual here.
I figured out I was bisexual in my early teens when I fell for a girl. Personally, I never had a problem with it - I had few friends of my own age and most of my friends were adult, bisexual women. I also came from a very liberal family, I was brought up as a liberal atheist and basically I skipped the self-hating stage of sexuality.
The part I feared was coming out to my family. Until two months ago I spent my life in tiny rural villages and towns in Nowhere, England. I never mentioned my bisexuality in public, not from fear of the consequences - I still think the reaction would be muted - but because there was little interest in alternative sexuality. Frankly I only cared about my family's opinion anyway. Interestingly enough, I came out to the members of my family on separate occasions, in order from least to most intimidating ...
I came out to my little brother perhaps a year or two after I realised. He would have been in his early teens at the time and we'd always had a good relationship. The specific impetus was that my brother had made some moderately homophobic comments which I knew (1) he didn't mean and (2) he hadn't thought through - something about not minding homosexuals but not wanting one in the family. So I told him I was bisexual and we talked about sexuality and tolerance there and then. I've never, ever had a single problem with my brother about this; we often tease each other about girlfriends, talk about women etc.
I waited some time for the right moment to come out to my parents. I came out to my mother first, then my father. I think on both occasions I was alone with my parent, we were already discussing sexuality and I seized the opportunity. Bear in mind that my family is liberal - when your parent has just said that they wouldn't be upset if any of their children were homosexual, it's a lot easier to confess!
I do remember clearly that my mother suggested it was a phase. I was offended and told her that (1) I'd known for years and (2) she wouldn't say that if I'd said I was straight. My mother is now training as a social worker so back then she recognised a heteronormative comment when I pointed it out.
In short, while the social support in Nowhere, England was zero, I benefited hugely from the fact that both I and my family believe in equality regardless of gender, race and sexual orientation. I've had the occasional tense moment with various members of my family but I never had to fear losing my good relationships with them.
PS: I also have an older brother whom I don't get along with. I never mentioned it to him and I don't plan to; I have as little interest in telling him intimate parts of myself as he has in learning them.
PPS: As far as I know no member of my family mentioned it to another until after I'd finished coming out to all of them. Either that or they're devious people and I never knew.
PPPS: I now live in Bristol where I'm a happy member of my university's lesbian, gay, bisexual & transgender society.