The Old City can only be navigated on foot — all traffic stops at the gates. It's smaller than it feels, about a square kilometre; give yourself time to get lost in the alleys.
For the rest of the city, the light rail (one line along Jaffa Road) links Mount Herzl in the west to the northern suburbs and is the single most useful piece of transport — stops at the Central Bus Station, the Machane Yehuda market, Jaffa Gate (Damascus Gate) and on. Buses cover everywhere else; neither buses nor the light rail run on Shabbat (Friday sunset to Saturday nightfall).
Taxis are plentiful; Gett is the local app. Driving in the centre is unpleasant — parking is scarce and one-way systems are labyrinthine.