I feel your pain. When I first came out, everyone was fine with it, but I had a hard time coming to terms with it. I felt like I wasn't 'queer enough'. I felt like I just wanted to be either one or the other. Now I'm okay with it. And when people ask me how I define my sexuality, it just depends on how I'm feeling that day. Sometimes I say I'm queer; other times I say I'm gay; I don't really use the term 'bisexual' anymore, but I usually use the term 'pansexual'. Basically, I like people. All kinds of 'em.
How I used to describe bisexuality to other people is, I am attracted to both sexes. You will probably run into people along the way - queer and straight - who will tell you that bisexuality is not 'real'. Screw them.
Good luck on your journey.