You know, I always see these threads about discrimination against furries or people's friends not accepting someone because they are furry, and the only thing I ever know how to respond with is "being furry has a social price tag, just like everything else." Just like Pilose said, things come with social price tags, (such as tattoos, piercings, hell, even down to your hair color) and you're just going to learn how to deal with it. By this I mean, there's just no denying that being furry is weird, but I mean, weird is not always a bad thing. It's just like if you decided to dye your hair some outrageous color and walk out in public -- It's weird, not necessarily wrong -- but you better be prepared to get some rude or curious reactions. That's just how people are.
Really all you can do is answer the questions people throw at you, and deal with their comments. The worst thing you can do is get mad and start defending it intensely, because it's no different than trolling online, some people make fun of you and do it just for a reaction, and when you respond, you're doing what they want. I guess what I'm trying to say is that you can't let them get to you. The furry fandom is just like every other fandom, even down to sci-fi and anime; there are people mildly interested all the way to people who have fetishes based around the fandom, I just think the furry fandom catches the most hell because they get so defensive over it.
But, my point is, you can't just run around being a furry and not expect negative or odd reactions because it is not a normal every day thing.