I've said it in other sections so I'll say it again here - you can explore the city centre on foot or by bike and these really are often the best options. If you're walking, sensible shoes are wise over the often cobbled streets. Bike hire is available from almost every hotel in town (typically around 140 krone per day), although you'll find much cheaper deals from specialists in the centre, such as Rent-a-bike (Gothersgade 157; www.cycleborsen.dk; +45 3314 0717), which charges 75 krone for a day, 165 krone for three days and 285 krone for a week.
When you do use the public transport system you'll find it first rate, efficient, inexpensive and relatively simple to use.
Buses and trains
Tickets for buses, the small metro network and suburban (S-Train or S-Tog) trains are integrated and based on a zoning system (travel in one zone costs around 20 krone, half for children and permits travel for an hour). The network gets going around 5am and runs through to half past midnight. The free maps in the tourist office (www.visitcopenhagen.dk) show the bus routes and stops. Don't forget to stamp your ticket to validate it before you travel.
You can flag taxis on the street if their sign is lit. The minimum fare is around 20 krone plus about 10 krone per kilometre. To order cabs try Taxamotor (+45 3810 1010) or Codan Taxi (+45 7025 2525).
The Flying Fish (www.findvej.dk; +45 32 964 964) water taxis operate in the summer and you can hail them from the shore (whether they spot you is another matter). You can book ahead and it can be a great way to get around if you need to cross the harbour. A one-way trip will cost about 100 krone.
Getting to the city
For information about travelling from the airport to the city see my Copenhagen flights page.