Rosario is a large city but the main attractions are concentrated enough that you can cover them on foot if you are based in the centre. The walk from the Monumento a la Bandera to Parque de la Independencia and back takes less than an hour, passing most of the key sights.
The city has an extensive bus network that covers the outer neighbourhoods and the waterfront. Buses require a rechargeable card (tarjeta Mio) rather than cash. Taxis and app-based remises (similar to Uber) are widely available and cheap by European standards.
Cycling has grown significantly as a transport option — there are bike share stations around the centre and a growing network of dedicated lanes, particularly along the Costanera waterfront. This is one of the better Argentine cities to explore by bicycle.