Lets be honest. Not all new buildings are good architecture. And even buildings like the Operahouse, creates a lot of debate. But in the last 20 years the building cranes in Copenhagen have risen to the sky. Ørestad is a new part of Copenhagen with buildings by Bjarke Ingels like Bjerget (The mountain) and 8tallet (Number 8). And in Nordhavn they are building a whole new part, with new houses. Some are good, and some bad. Also the Concert hall in DR (Danish Radio and TV) Byen by Jean Nouvel is pretty nice next to the IT Univercity by Henning Larsen. Here is also the round Tietgen kollegium by Lundgaard & Tranberg

Opposite Tivoli you will find Copenhagen Towers by Lundgaard and Tranberg. And a third building by the couple is the Playhouse (Skuespilhuset), the new royal Theater by Nyhavn. The harbour is dominated by the Opera House by Henning Larsen. There are a lot of debat about. Is it too big? Looks like a Mercedes Grill, and what not.
And then you can go to Christiania, to see all the small self built houses, and it is more art than architecture.

Nice City spaces includes The school yard / Play ground at Israels Plads – Opposite Torvehallerne -, The Red Square and Black square in Outer Nørrebro is designed by Bjarke Ingels Group, and is a must see if you have the passion. Also the Assistensens Churchyard is worth visiting. You get a bit of the harbour atmosphere at Havnegade.
In Vesterbro you can see the Sønder Boulevard, that has been transformed from a four lane street, and into a very cosy hipster friendly area. You can also visit the Childrens playground at Skydebanehaven.
On Islands Brygge, you find the garden by the Harbour Pools, and if it is a warm and sunny day, hundreds will go here, swimming and sun bathing. The Ofelia Square just by the Play House Theater is also a good spot, though it can be quite windy.