Hi Helen,
My suggestion would be to try an entry-level hardtail mountain bike. Trek and Gary Fisher models start at about $320. Mountain bikes are generally easier to ride and more comfortable than road bikes. And you sort of have to worry less about them. For example, you can get a flat on a road bike just by riding over a 1" rock. It's true that road bikes are more comfortable over long distances, but 20-40 miles of road riding on a mtb is very doable. Usually the longest I'd do was about 50 miles. Beyond that, I'd consider a road bike, a mtb with slick tires, or a hybrid.
If you have your heart set on a road bike, I highly recommend the Trek 1000, which is $710. One of the guys I ride with got it a few months ago and he's very happy. It's a very good entry-level road bike. In your case I suggest you consider the Trek 1000 WSD (Women's Specific Design), since you mentioned that you have short arms. But by all means, try both. Usually women's specific bikes have a shorter top tube, which makes it easier for people with short arms or short torsos to comfortably reach the handlebars.
At 5'7", you would probably fit well on a size 51 cm, but the only way to know for sure is to get your inseam measured. They should do this at the bike shop by getting you to stand barefoot against a wall, with your feet about 6" apart. Then you'll hold a binder or hardcover book between your thighs and push it up as high as it will go -- simulating your position on the bike seat. The salesperson will then measure the distance from the top of the binder straight down to the floor (usually in centimeters, since road bike sizes are in cm) and that's your inseam. Then you take that number and multiply by .667 and that's a good approximation of your bike size. So for example if your inseam is 77cm, then 77cm * .667 = 51.4 cm. Your bike size is about 51cm.
Sorry for the long post, but I hope it helps. No matter what bike you decide to get, it's important that it fits well so that you can enjoy it to its full potential.
~Cari