Try and catch a world record Lake Trout! Below the dam, the green river is very spectacular, carving though a deep canyon that has good trails along the bottom. Be careful of going out on sandbars or crossing low parts of the river, however, because if the dam increases flow to generate increased power, the water level can change dramatically in a matter of minutes, and strand you, or make you swim in dangerously cold water.
You will be just north of the Vernal area- Dinosaur Land, some interesting desert canyons and such if you come down from the high country.
Farther south (thinking about 5 or more hours) is the San Rafeal, Arches, and Canyonlands area.