Visiting Turkey as a globe traveler would surely bring you to Istanbul. Spending many days according to your taste you will be wondering what the rest of the country is like.
From Izmir to Ephesus
After you land in Izmir Airport you can find many Rent a Car options at the airport. If you wish to take public transportation there are busses that runs to the City Center (Konak area) and there you can take another bus or a taxi to the main bus station to find yourself a bus that goes to the town of Selcuk where ancient Ephesus is located. This option might take about 3 hours. From there you might need to walk or use taxi between the places of interest. Or you basically rent a taxi from the airport to take you to Ephesus, which might be an expensive option.
There is always an option of a professional touch to this excursion. You can take a look at this tour option from this link.
Ps: If you like animals, bring some cat food for the todays residences of Ephesus.