Everyone says it's best to start new shooters on a .22, and maybe it's true. But it isn't necessary; a person who's never fired a handgun before will get good with any normal handgun round with a bit of practice. .500S&W, or .454 Casull, might not be such great choices, but 9mm/.40/.45/.38/.357 are fine. There are light enough loadings in all of these rounds that a new shooter shouldn't have trouble, unless he/she is extremely recoil sensative (if you find the recoil of a downloaded .38 out of a 4" K-frame to be very unpleasant, I'm not sure how to help, really).
My first and second handguns were .45 Automatics. My third was a .45 Colt sixgun. So it's doable.
When I bought a revolver for a female friend of mine (she's built like a ballerina), I didn't get her a .22. It was to be her ONLY handgun, since we're both college students and money is tight (well, for her, I had plenty, heh). So I went looking for a used .357 revolver (since she already had a .357 magnum lever carbine), and I found her a Smith & Wesson Model 28. That's a 4" N-Frame revolver in .357 magnum.
With .38 plinking ammo, the gun is a pussycat. She doesn't think it's too heavy, either, and the thin factory grips fit her small hands perfectly. She didn't complain of tigger reach problems, though she had to teach her finger how to do the double action trigger pull.
I'd recommend, then, a .22 only if you plan on getting her a second serious gun later, or if she'll buy one herself. If it's going to be her only gun, and you want it to be more useful for self defense, I'd say a 4" K or L framed revolver (or a Ruger Security Six, a used example of which can be had for $200 or so), in either .38 or .357. Plenty of different grips available for those with smaller hands, and there are plenty of "cowboy action" .38 loads out there that are positively anemic, but perfect for just plinking and target shooting.
Best of all, though, a used revolver can usually be had for $200-$350.
As for automatics...I've heard that the Kahr T9, being a 4", single stack 9mm, has a very short trigger reach and is good for female or young shooters. It also has only one control, the slide release (being double action only), so it's simple to operate. It'll cost you a good deal more than a used revolver, though.
