It will prove much simpler in person than it seems online — such is the nature of ground transport in Turkiye.
The easiest option for you would be to ask your hotelier to arrange for airport transfer in advance. They will offer several options, from the very expensive one where their driver or shuttle transports you, to a private taxi, a bus ticket, or train or metro.
In addition, every travel agent in Turkiye is licensed, and they all want to help sell you airport transfer. Choose one, contact them (WhatsApp is very useful for this), and allow them to help you.
To DIY: in Izmir, there is a metro station at the airport. From baggage claim, one simply crosses the street and follows the signs. Purchase a bilete (ticket) at the booth or at the automated machine. You will need Turkish lira in cash.
You could also take the train, TCDD.gov.tr is the website for Turkish Railways.
If you prefer to take a bus, Flixbus or Rome2Rio websites can provide you with bus schedules, and it is simple enough to purchase those tickets when you arrive. Havabus and KamilKoc are local bus companies.
If you are new to Turkiye and not fluent in Turkish, I highly recommend that you request assistance in booking transfer from your hotelier or from a travel agent in Turkiye. They are wonderfully kind and helpful.
