There are large regional networks of public transport:
Nordhessischer VerkehrsVerbund NVV
Rhein-Main-Verkehrsverbund RMV
"Hessenticket" for day-long on Local trains in Hesse and Mainz for up to 5 people.
German National Railways offer the "Schönes-Wochenende-Ticket" for day-long unlimited travel on local trains nationwide on week ends.