Lisbon is a small city and much of it is walkable, but the hills are steep and the cobblestones uneven. A Viva Viagem card, loaded with credit, covers the metro, buses, trams, funiculars, and the Santa Justa Lift — it's worth buying from any metro station on arrival.
The metro (four lines) is the fastest way to move across the city, though it doesn't serve the Alfama or Belém. Trams are the romantic option: line 28 cuts through the Alfama and Graça on a spectacular route, though it's always crowded with tourists. The three funiculars — Glória, Bica, and Lavra — provide useful shortcuts up the steepest hills and are included on the transit card.
Buses cover the whole city including Belém (tram 15E or bus 714 from Praça do Comércio). Taxis are metered and reasonable; Uber and Bolt operate widely and are usually cheaper for longer journeys. The city is flat enough along the waterfront that cycling is pleasant there, though cycling in the hillside districts is not for the faint-hearted.
Ferries from Cais do Sodré cross the Tagus to Cacilhas, Montijo, and other points on the south bank — a cheap and scenic way to see the river.