Katowice is around 200 kilometers away from Zlin. If you are landing there, you can get to Zlin by bus or train. Once you get to the airport, you have to first reach the Bus or Train Station.
Full schedules for flights, as well as other essential travel information, can be found at the user-friendly www.katowice-airport.com.
There is Flixbus going to Zlin from Katowice Bus Station, once per day at 14:10. The prices may vary from 10 EUR to 20 EUR. For more recent information, actual schedules, prices and booking your ticket, please check the website of the company: https://global.flixbus.com/
You can also get to Zlin from Katowice by train. However, more information can be found if you contact directly the Polish or Czech Railways.
Please, check the website for further information and how to contact: https://www.cd.cz/en/