There are many ways to go to Selcuk but the most useful ways is using train in İzmir to go Selcuk. You can search suitable train times for you and arrange yourself according to this times because there is no train every time. I will share train times and a link that you can also check from official site.
After landing in Izmir Adnan Menders Airport, you can directly go to train part and ask your ticket to the Selcuk train station from airpot. The road takes nearly 2 hours with train but it is the most useful option to go to Selcuk from Adnan Menderes Airport.
There are 8 trains in a day and the prices is really cheap. Here is an example of train hours and prices to go to Selcuk.
The currency is 18 Turkish Liras is nearly 1.11 euros and 1.16 Dollar. You can get ticket from the ticket personel from airport and to go to back to Izmir again you can get the ticket from same station.
The ticket seller coming 20 minutes ago to sell tickets before the train came the station. You can wait in the queue with other.
Have a nice trips!