First, the basics. You want a bike with a triple chainring (the gears attached to the pedal) and a seven- or eight-speed cassette (the gears on the back wheel). Those won't actually give you 21 or 24 usable speeds, as some gear combinations don't work well. But you'll still have 15 to 20 effective gears, plus a lot more on the low end for going uphill. For tires, you want slick (no tread) tires, offering less rolling resistance and more speed on pavement. The handlebars should be nice and upright, so you're not leaning over and putting a lot of weight on them. You can add more cushioning, but the better solution is to wear a pair of inexpensive bike gloves. As for saddles, all sorts of options exist, including good women-specific models as well as super-cushioned ones.
A lot of bikes fit this general profile in the $250 to $400 range. K2's Rosario ($350; www.k2bikes.com) is typical. It's designed for comfort, with gel-padded grips, a wide saddle, an upright riding position, and 24 speeds. A nice bike for city riding. The Novara Metro ($300), available at REI stores, has all the above plus a seat that rides on a spring suspension and suspended front wheels to really absorb shocks. Finally, there's the Fuji Regis ($280; www.fujibikes.com), which doesn't have suspension but still offers a very comfortable ride.
You could roll any of these bikes straight out of the store and be happy. But talk with a bike salesperson about saddle or tire optionsyou might be able to swap out some components to suit your personal tastes.