(...) take a "schoenes wochenende ticket" or a similar one for in the week (...) but you can use it on all non IC/ICE trains for a whole day (...)
This aspect can't be stressed enough. DB conductors can be pretty stern about this, and make you either leave the ICE/IC immediately or pay an extra fee. If's there an "RE" in front of the train code on the display, you're good to go.
DB actually offers a free and pretty useful app to help you with your journey:
DB Bahn: DB Navigator