Train travel is fast and convenient! Train travel is cheap and comfortable! And above all: train travel is FUN! In Denmark, all these statements are true, and Copenhagen happens to be a hub for train travel, whether you're exploring this little country, or planning to visit other parts of Europe.
Just how convenient train travel is here becomes apparent right after landing at Kastrup airport (Copenhagen International). 21 kroner (less than $3) and 20 minutes later, you're at the Copenhagen Central Station. Or if you choose, you can go the other way, and in 30 minutes wander the streets of lovely Malmö in Sweden. Local trains run very frequently here: some as often as every 5 minutes but always at least once an hour. They span a dense network around the greater Copenhagen area, making travel effortless.
For travel to further parts of Denmark or Scandinavia, InterCity trains are the most excellent way to travel. You can reserve your seats in advance in either smoking or non-smoking cabins, or even the low-noise or rest wagons (no cell phones or loud noises allowed in those). Trains are quick and very clean and zoom at comfortable speeds of about 100 mph. Bread, coffee, tea, juice and bottled water are provided at no charge, but a large selection of snacks and drinks are available for purchase. Travel time to anywhere in Denmark does not exceed 6 hours, due to the small size of the country.
Copenhagen's main train station is located very conveniently right at the center of the city. Several hotels are located adjacent to the station, among them Hotel Astoria, Golden Tulip Imperial Hotel and Sofitel Plaza Copenhagen. Many tourist attractions are also nearby: Tivoli gardens are located right behind the station, Hard Rock Cafe is one block over and Stroget, Copenhagen's famous pedestrian-only street is a 5-minute walk from the station.
Copenhagen Card allows free and unlimited travel on all local trains around Copenhagen with the exception of Copenhagen Central Station (København H St.) - Airport (Københavns Lufthavn Kastrup).