Most people fly into İzmir and leave for the beach or Ephesus the same afternoon. That's a mistake. İzmir is Turkey's third city and was Smyrna for most of its history: an Ionian town that claims Homer, a Roman city rebuilt by Marcus Aurelius after an earthquake, and for three centuries the busiest port of the Ottoman Levant, where Turks, Greeks, Armenians, Jews and European merchant families lived side by side. The Ottomans called it Gavur İzmir, "infidel İzmir", for how foreign it looked. Most of that city burned in September 1922, and what you walk through today is what came after, built on top of what survived.
It's also the easiest big city in Turkey to spend time in. Life happens outdoors and by the water: people fish off the Kordon promenade, sit on the grass at sunset, cross the gulf on ferries for the price of a bus ticket and insist, against the rest of Turkey, that a sesame bread ring is a gevrek and not a simit. The pace is slower than Istanbul's, and five days lets you settle into it.
This plan keeps each day in one part of the city. Day one is the old center, the bazaar and the Kordon, day two Kadifekale, Jewish İzmir and the Asansör, day three Alsancak and Kültürpark, day four a trip to Ephesus and Şirince, and day five the far side of the gulf. If you have longer, the best trips out of the city are after day five. İzmir is a good way into Turkey as a whole, too: the food, the tea and the rituals are all here, with a little more sea air.