Sounds like a toss-up between a mountain bike or a hybrid. Hybrid , if more riding on paved streets, mountain if not. Like mentioned above, swapping the knobbie mountain bike tires to a smoother & faster riding tire is no problem. I do that on my mountai bike .
http://www.giant-bicycles.com/en-us/bikes/women/ Look over this site at their mountain & lifestyle [ what many of us call hybrids ]. I like Giant bikes & can recommend them, IF you have a dealer close by. $300-500 is a lot of money to many,, including me,, but it won't go far in a bike shop. Fortunately, Giant has a good many bikes that are well priced. The right type bike & the right fit, for your body size, is important !