Bargarh town itself is small enough to cover on foot; the bus stand, market, and main streets are within easy walking distance of the central hotels. Auto-rickshaws are available for longer runs within town and will take you to the outskirts for a negotiated fare.
Getting to the sights around the district requires more planning. Shared tempos (three-wheeled passenger vehicles) and local buses connect Bargarh to Barpali (about 30 km south) several times a day, and this is the easiest trip to make without a private vehicle. For Debrigarh Wildlife Sanctuary you need to reach the Dhodrokusum gate, best done via Sambalpur; the sanctuary itself is traversed by jeep on forest tracks, and vehicles can be hired at the gate or arranged through the forest department. Nrusimhanath temple on the Gandhamardhan hills is a long half-day drive of around 165 km and is most easily visited as part of an overnight trip or with a hired car from Sambalpur.
For independent travellers without their own vehicle, hiring a car with driver from Bargarh or Sambalpur for a day or two is the most practical way to reach the outlying sights. Rates are reasonable by Indian standards and most hotels can arrange this.