DSB has a cheap ticket type called Orange Sviptur for day trips within a region. A ticket is valid for an afternoon from 4pm to 4am and lets you travel out and back on the same ticket. However on Saturdays tickets are valid all days.
This makes the ticket great for a Saturday exploring our region – or combining it with other regions and see other parts of Denmark. The tickets does not cover the Storebælt bridge but you could get off the train and buy a single ticket for this part. That way you can go for a day trip to Copenhagen and back for just 278dkk (plus the ticket to cross the bridge).
Tickets can be bought on the day either in the ticket machines or by SMS. To buy a ticket through SMS you need to have a mobile subscription that lets you buy services through the phone. E.g. send a SMS to 1412 with the content ‘orange syd voksen’ for the South region.