There is public transit from Dulles into downtown D.C., but it takes awhile. Go to the Washington, D.C. forum and you'll see a post in the "top questions" box about how to get from Dulles into D.C. using public transit. From there you can take the train to Fredricksburg. If you time the trains right, best case I think it would be about 2.5 or 3 hours from Dulles to Fredricksburg via that route.
You should strongly consider renting a car and driving. That is the best all-around option balancing cost, time, and convenience.