Szczecin is large enough that you'll want public transport for some of it but compact enough in the centre that the main sights are walkable. The old town, castle, Wały Chrobrego embankment, and cathedral are all within comfortable walking distance of each other.
The tram network is the main public transport option — it covers the city well and is cheap. Tickets can be bought at machines or on the tram. Taxis are available and not expensive; local apps work here. Cycling is increasingly well-served with dedicated lanes along the riverfront and through the park system.
For the Oder riverfront west of the centre, a boat or water taxi is the most atmospheric way to explore — day trips on the Oder and into the surrounding lake district are available from the city waterfront in summer.