Tampere is compact enough that most of the central sights are walkable from the railway station. The tram network covers the main corridors — Hämeenkatu is the central line, and extensions run to Pyynikki and Hervanta. Single-trip tram tickets can be bought on the tram; day passes are better value if you are moving around frequently.
The city is well set up for cycling with dedicated lanes on most major streets. City bikes (Tampere Fillarit) are available from multiple docking stations and run from late spring to early autumn. For Pispala and Särkänniemi, a tram or bus is useful — the terrain is hilly enough that walking there and back repeatedly gets tiring.