The city centre is compact and easily walked end-to-end in twenty minutes. The NET tram (Nottingham Express Transit) runs two lines through the centre and out to the suburbs, including the universities and Clifton; flat fares, contactless ticketing. Nottingham City Transport's buses are extensive but slower; the network is well-rated and runs late into the night.
If you're going west to Wollaton Hall or south of the river, the bus or the tram is faster than walking. Cycling is workable in the city — the Big Track along the River Trent is the obvious leisure route — but Nottingham's hills mean it's not the natural choice for general transport.