The old town and the harbour are easily walkable. Beyond that, the city's blue-and-white dolmuş minibuses run along the seafront roads in both directions and to the beaches and outer neighbourhoods. Hail them anywhere on the route; fares are low and paid in cash.
For Ephesus, Selçuk, and other day-trip sites, the easiest options are an organised tour (sold along every other shopfront in the Kaleiçi) or a private taxi for the day. Public dolmuş minibuses run to Selçuk from the otogar every twenty minutes and from there local buses continue to Ephesus's lower gate.
Renting a car gives the most flexibility for the wider archaeological loop south to Priene, Miletus, and Didyma, or inland to Pamukkale. Several international rental chains have offices on the seafront.