Time tables can be found at the Railway Station or you can plan your journey at www.rejseplanen.dk (switch to English or German in the top right corner)
DSB has a number of discount schemes. If you are a student (or below 26) you can get a “Wild Card” giving you up to a 50% discount to train rides. For further information see here (in Danish).
You can also buy tickets called “Orange”, which are cheaper train tickets, but they must be booked well in advance. Read more about this here (in Danish).
Be aware that you will need to buy a ticket before boarding the train or you risk a fine.
For both buses and trains you can buy single tickets, multi-ride tickets or a monthly travel card.
If you are going on a longer trip, it can be a good idea to make a seat reservation as trains can often be crowded, especially on weekends.