A fast otoyol (expressway) goes from İzmir all the way to Alaçatı and Çeşme, making it possible to drive the distance in less than an hour. The Alaçatı exit is the last one before the end of the highway at Çeşme. (Here's how to pay the toll.)
Çeşme's bus terminal (Otogar) is inland a few blocks from the bay on the İzmir road, about a 15- to 20-minute walk from the main square or the harbor (Liman). If your baggage is really heavy, you can take a Çeşme-Ilıca-Alaçatı minibus from the lot just south of the Tourism Information Office to the Otogar.
Bus transportation via the buses of the Çeşme Seyahatcompany between Çeşme's otogar and the Uçkuyular bus terminal south dof the center of İzmir (map) is fast and frequent (every 15 minutes from 6 am to 6 pm), so you can pass through easily enough. Most of the buses continue to from Uçkuyular to İzmir's Büyük Otogar(main bus terminal), from which it's easy to catch onward buses north to Bergama or south to Selçuk/Ephesus, Kuşadası, Bodrum, and other destinations.
Ferryboats arrive in Çeşme from the Greek islands and dock at the Liman (harbor), easily visible across the bay from the main square. It's only about a 15-minute walk around the bay from the square to the Liman.
Distances & Travel Times
Istanbul: 700 km (435 miles), 8 hours
İzmir: 85 km (53 miles), 50 to 75 minutes
İzmir's Adnan Menders Airport: 90 km (56 miles) SE, 60 to 85 minutes by car, 2+ hours by bus
Selçuk (Ephesus): 125 km (78 miles), 2 hours by car, 3 hours by bus