When asked in past years what to wear while training with weights, I used to recommend running shoes. They had a built-in arch support that kept feet from facing compression injuries during heavy leg and back work, and they had a good tread which stuck to a calf block when doing standing or seated calf raises for the lower legs. Additionally, they didn't cost that much (under $100 per pair) and they were quite sturdy and trendy looking.
With the recent advent of cross-training shoes, however, I have now changed my mind. Cross-trainers have been developed specifically with bodybuilding
workouts in mind; both the type in the gym and the type on a bike, stair climber or treadmill that are designed to help burn bodyfat when you are in a peaking cycle. Cross-trainers even have great built-in lateral stability, making them ideal for organized aerobics classes, you know, the type where you jump along and spontaneously shriek at the top of your lungs from time to time to show how enthusiastic you are about the burn.
Cross-trainers tend to cost a little more than running shoes; at least the range of prices isn’t as great as with shoes specifically designed for running, but they tend to be even more colorful and trendier than joggers.
But then you will probably only need one pair of cross-trainers to last you all year, unless you happen to wear them all the time.
If you can afford the price of a podiatrist's appointment, I'd also suggest you have your feet examined. Many women (and men, too) have slight irregularities in the construction of their feet or the length of their lower limbs, which can be corrected quite simply by wearing an orthotic device in the cross-trainers. I know one woman at the gym where I train who cured a long-standing lower-back injury with a pair of orthotics!