The new Erkilet airport is connected to the major Turkish cities and some major European cities and awailable to fly here everyday in various times.
check the websites for awailable flights below listed airway companies:
-THY Turkish airlies,
Theres a train network around Kayseri.The main station connected to south (Adana) east west by TCDD trains. To Adana you can take the railbus also for 10$ you can go to south for example in 4 hours.Also you can take trains to Ankara and Istanbul from the main station.
It will be cheepest and interesting way to see the city center by tram have a look for the tram stops of "kayseray" tram...
By coach buses you can reach everywhere in Turkey from Kayseri.The Bus terminal is located at citycenter.