Well, for your Grandma, show her a fursuit or someone in cosplay. Show her a lot of it. People wearing them in full dress, and tell her THAT is what you intend to do. Make it plain as day, promise her, and do all that nice junk to prove you're not going to go prancing around like a jubilant doe wearing nothing but false extremities. Really, that's all you can do. If she refuses to understand it at that point... Well, that's crap, but she obviously won't listen to reason if so.
As for your mother, tell her you are really hurt that she would be so condescending of your simple interest in the furry fandom and anthropomorphic animals. I mean, Christ, ask her if she likes Looney Toons or Tom and Jerry, or hell, even Thundercats or some show that she may have grown up with featuring anthropomorphism or humans with animal-like characteristics. Then tell her that you just think that they're really awesome, or something simple that is not misleading or suspicious, and that you made a "cartoon character" to represent who you are. But most importantly, tell her that she has really hurt your feelings that she is so unwilling to accept something you consider a personal interest. No matter the case, iterate and reiterate the point you intend nothing venereal (if they so assume you are being such) or obscene with it.
A question: is your family a devout religious one, or are they all across the board in regards to belief? If they're extremely devout, that might be a part of the problem. Most religions place the animal existence below the human, so they may think you're trying to lessen your being by dressing up as an animal, or something like that. If so, you'd have to fight not only misconceptions, but strongly held beliefs. Be careful while treading there. I really hope that isn't the case.