When I first got into firearms a few years ago, I thought the same thing. "A .22? Oh, please, might as well just shoot a kiddie BB-gun!"
Then I actually tried one. My mind changed very quickly.
Shortly thereafter, a 10/22 somehow followed me home. I affectionately call it, "The Assault .22" because, of course, it's "capable of accepting a High Capacity Clip
This rifle is just plain fun to shoot, and it's tough to beat one for harvesting small game like rabbits and such. Ammo for it is dirt-cheap, too. I cannot think of a better rifle to use for practicing marksmanship fundamentals than a .22LR of some sort.
So, how do you deal with the "stigma"? Whoever doesn't like your choice of firearm is free to purchase a so-called "real gun" in a so-called "real caliber" for you with their money if they think it's so important. Anybody who opens their mouths at you like that, I suggest you consider mentioning that option to 'em. Oh, and they can buy your "real caliber" ammo for you, too. Until they do, just keep thinking of how much fun your .22 is to shoot, and how much easier on the wallet it is than their so-called "real guns".