The mountains around Hiawassee make for excellent day trip territory. A short drive south on US-76 brings you to the trailhead parking for Brasstown Bald — worth doing as a half-day excursion in itself. The Appalachian Trail crosses the area: Blood Mountain, a popular summit on the trail, is about an hour's drive south and offers a strenuous but rewarding hike.
The nearby town of Helen, about 25 miles south, is a quirky Bavarian-themed village on the Chattahoochee River with shops, restaurants, and the Oktoberfest that runs from mid-September through November. Equally worth a detour is Tallulah Gorge State Park, about 40 miles south — one of the deepest gorges in the eastern United States, with dramatic rim trails and suspension bridge.
North Carolina is just over the lake. Murphy, NC, is the nearest town across the state line and has a modest historic district. For a longer excursion, the Great Smoky Mountains National Park is two to three hours northeast — the most visited national park in the United States.