@TLD; I visited Zlin with public transport from Brno in 2018. I don't remember how exactly, but I don't recall too much complications.
I will look into the zoos you mentioned.
I'l probably be using public transportation for this trip. I do have a driving licence so renting a car is a possibilty, it's really just a matter of preferences. I really dislike driving.
I went to Zoo Zajezd (which is really exceptional if you like chameleons) by public transport from central Prague yesterday, so it is doable. I took the 26 tram to Nádraží Veleslavín from Prague Main Station, then the 342 bus to Buštěhrad/Bouchalka, from which stop it is a 15 minute walk or so to the zoo. I then got a Bolt taxi back to Prague Zoo from there, but it would have been possible to return via public transport. In total, it took about an hour and 20 mins.