There is a lot of water around the city. You can swim in the harbour, the water is clean. Not very many places in the world where you can do that. : – ) Please note that officially it is forbidden to swim, where there are no life guards – but nobody really cares. The official harbour pools are at Islands Brygge, and at Fisketorvet. The one at islands Brygge is best for kids, the one at Fisketorvet is not so crowded.
If you are more into sand between your toes, you can bike or go by Metro line (towards the airport) to Amager Beach. It is low water which is perfect for small kids, and you have a view of the Øresund Bridge, the Windmills, Malmø in Sweden at the other side of the sound, passing ships, and the Airport. Lots of entertainment on this beach!!!
There are also a beaches if you go north or south of the city, along the Øresund coast.
For indoor swimming there are some nice areas: the best for kids is at DGI city, just next to the Central Station. Øbro Badet next to the national soccer Stadium in Østerbro, Parken, is really beautifull, builth in the 1930´s.
There is a big network of bike routes across the city, making Copenhagen the best city for cycling in the world. And the network is still growing!!!
You can rent bikes at the hotels or at the biggest operator in town http://copenhagenbicycles.dk
for guided tours, these are the best: http://www.cocutours.dk
Skating and skateboarding
Theoretically you can skate all over the city, but please note that the police dont like that you skate on the bike tracks. In Fælledeparken there is the best outdoor skate park in town, where you can go with your board. At The Red Square, there is a a lot of local boys hanging out on there home build ramps.
For indoor skating you can go to Enghave, but it cost 50 kr to enter. If you dont mind a little smoke, go to the indoor Wonderland skate park at Christiania, which is free. https://skateparks.dk/kobenhavn/copenhagen-skatepark/ (Only in danish)
Every other friday there is a Friday Night Skate (FNSCPH). It is only in the summer, and is a mix of lycra inliners and side by side hippies and punks 😉
Go out to the old ship yard at Refshaleøen opposite the Little Mermaid. You can bike or take the harbour buses. The are more and more cafes and bars popping up, houseboats and old industrial decay. Every second sunday, there is a huge Flea market in one of the halls. You can also go swimming here.