The quickest way to get to the Aegean Coast is by plane. There are daily flights leaving Istanbul to airports in Izmir and Milas/Bodrum. There are direct flights to Izmir and Bodrum from Turkey's other major airport hubs: Ankara, Adana, Trabzon, Dalaman, and Antalya.
There are a number of bus lines serving the Aegean coast. The ride from Istanbul to Izmir takes around 12 hours, and from Istanbul to Bodrum takes 16.
Having your own car will save you a lot of time between sights and allow you to explore some of the neat places that are inaccessible by bus. There are car rental agencies in all of the major towns, but be sure to check their conditions as some have more liberal drop-off policies than others.