Depends on what you're into. If you want to see architecture and history, go to Prague for 2/3 days. If you want to see what life is actually like in the Czech Republic, skip Prague and find a small resort/camp site somewhere in south Bohemia, maybe near Sumava national park (but there will be a wider mix of people, so you won't get to know so much about the area). You'll probably be able to meet some people who speak English and want to have a few beers with Americans. Also, for accommodation use airbnb. There is a service in Germany call mitfarhgegehen (I think it's called) where you can find people driving in certain directions willing to take others for some gas money. It's like electronic hitchhiking, so probably another good way to meet people (I've never used it though, and we don't have a similar one here in Czech). Good luck!
Logistically we're going to have to skip Prague in this venture but we'll must certainly look into the hitchhiking from Berlin to Copenhagen!