The old town and waterfront area around the Royal Museum, the State Mosque, and Jalan Trus are walkable, and recent improvements have added proper pedestrian paths through the main tourist areas. For most sightseeing in the centre, your feet are sufficient.
Grab (Southeast Asia's dominant ride-hailing app) works well in JB and is the most straightforward way to get around. Fares are low by Singapore standards. Regular taxis exist but drivers sometimes prefer to quote a flat fare rather than use the meter; settle the price before you get in.
City buses are cheap but infrequent and routes are not always obvious to visitors. For anything beyond the immediate city centre — Johor Premium Outlets, Danga Bay, the western beaches — a Grab or rented car is more practical.
Car hire is available at Senai Airport and from several operators in the city. Driving around JB itself is not always pleasant given traffic, but a car makes sense if you are planning excursions to Desaru, Pulau Sibu, or Endau-Rompin.