Williston is a car town, full stop. The city spread rapidly during the oil boom and the result is a sprawl of strip development along major arterials — walkable it is not, and there is no meaningful public transit to speak of. If you are arriving by air or train, renting a car at the airport is essentially mandatory. All the major rental agencies operate there.
Distances to the key historic sites are manageable on a day-trip basis: Fort Union Trading Post is about 40 km northwest of town via US-2 and a state road, Fort Buford is a few kilometres further on. The roads are good in summer; in winter or spring thaw, watch for soft shoulders and drifting snow on the smaller county roads.
Downtown Williston is compact enough to walk between the handful of restaurants and bars that cluster there. The Welcome to Williston visitor centre on Second Avenue West is a useful first stop — staff are genuinely helpful for maps, road conditions, and current hours at the historic sites.