Do I need to buy special visas to visit either of these countries? I'm an American and though just having my passport would be enough.
If US citizen no visa needed, passport is all you need.
If you are not staying more than 90 days in Schengen area which Czech Republic and Poland are parts of then all what you need is your passport. I read that Czech Republic (I don't know about Poland) requires proof of health insurance but they never asked me for it neither I heard that anybody else was asked for it.
Me to had the same doubt. Thanks for the clarification.
I think that there is no problem to travel for you. For Poland you can contact U.S. Embassy Warsaw
Have a nice travel!