Pretty much the only way to get to Tskhinvali at present is to travel north from Gori. The cheapest option is to take a marshrutka, which run multiple times per day between the two cities (although the routes are shut down for the time being, because of the hostilities between Russia and Georgia.
Tskinvali's city center is compact, and easily covered on foot.
Tskhinvali has long been the safest destination in South Ossetia to visit (although that emphatically does not mean that it has been safe), but it is currently unsafe for foreigners of any kind to visit. Even in better times corruption, violent banditry, and general lawlessness have made this an unappealing place to vacation.
While the long, bloody conflicts have just recently ended, the situation in the city is far from over. The health system is severely dilapidated in Tskhinvali; Basic medical supplies, plumbing, to name a few are very short in supply owing to a failing infrastructure that has been damaged severely during years of warfare. Be sure to bring the necessary medical equipment with you, and only drink bottled water.