The CBD, waterfront, Salamanca, and Battery Point are all walkable — the whole central area is barely a kilometre across, and the steep climbs are part of the charm. The city sprawls along the Derwent estuary beyond that, and you will want a car or transport to reach MONA, Mount Wellington, the Cascades, and the Royal Botanical Gardens.
Metro Tasmania runs the bus network. Buses converge on the Hobart Bus Mall on Elizabeth Street. Bus 49 climbs Mount Wellington as far as the Springs (not the summit). To reach MONA, the dedicated MONA ROMA ferry leaves from Brooke Street Pier — slow, theatrical, the proper way to arrive.
A car gives you Tasmania. Rental rates are reasonable and the roads are quiet outside of peak summer. For shorter trips, Uber and the local 13CABS service both work in the CBD.