Denmark has an excellent public transport infrastructure. It is therefore very easy to travel between various parts of the countries, even if you don’t have a car.
Due to the size of the country, air travel between major cities is not very popular. Trains, buses and ferries are very reliable and cover the whole country.
Most trains are operated by DSB, the Danish public railway company. Some local trains are operated however by other smaller companies. Trains travel very often between major cities, so it is easy to find one that is convenient to you. They are clean and comfortable. You can buy your tickets either online or from the machines in the stations, but they are not cheap however. You can use the DSB website to plan your train travel, but the search engine is available unfortunately only in Danish.
Bus travel is also popular, since it is usually cheaper than the train for the same route. The company called Abildskou is one of the biggest bus operators, connecting Copenhagen with many other cities. Their website is a good point to start searching for a bus route.
Ferries are important not only for helping you reach the islands, but also to shorten the travel time between various parts of the country. It is sometimes cheaper and faster to travel by ferry than to drive 200 km more. The biggest ferry companies are Færgen and Mols-Linen, but there are more than 20 companies in total.
As you see, there are many options for travelling around Denmark. You should consider all of them, comparing travel times and prices, in order to choose what suits your needs best.