Virginia City has no public transport connections. A car is essential. The nearest commercial airports are Bozeman Yellowstone International (about 90 miles north) and Idaho Falls Regional (about 100 miles west). Both have car rental counters.
From Bozeman, drive south on US-191 to Ennis, then east on MT-287. From the north via I-90, take the Belgrade exit to Bozeman and then head south. The drive from Bozeman takes roughly 90 minutes in good conditions. In winter, mountain passes can be treacherous — check road conditions before driving.
Virginia City is on US-287, 14 miles south of Ennis. The road from Ennis follows the Madison River the whole way and is very scenic. From Yellowstone's West Entrance, take US-287 north past Quake Lake (formed in the 1959 earthquake) and through Ennis — about two hours.