For $300 you may have some difficulty finding a "used" bike that she will be happy with. Visit a few reputable bike shops and tell them what you'd like. They can tell you what you need to know.
The other problem is getting a bike that "fits" your wife. If this is planned as a surprise, forget it. She'll have to go with you so the bike shop folks can make sure the bike fits.
BTW: fit is the most important feature when buying a bike. Brand, components, frame material, etc. are secondary considerations.
Best of luck to your wife in her aspirations.