You will need at youth hostel card. Buy it at the first youth hostel you get to. Hotels and Inns are fairly expensive in Denmark when you compare with Italian and French 1-star hotels. But then the standard is usually better. As with hotels everywhere you will get a better deal if you are two people sharing a room.
Some inns have an excellent reastaurant.
If you like to meet the locals you may want to ask the local tourist office for a list of private lodgings. Some people have a spare room which you may rent for a very reasonable price. Over breakfast you will get the chance to talk to the Danes at home. Bicycles and trains Denmark is a country of cyclists. Therefore it is fairly easy to get ones bicycle onto trains, busses and ferries.