Oh my…
I haven’t been to Norway to ages, but Copenhagen is really nice - and in June, to boot. So what hints I have to give are more related to Denmark/Copenhagen than to Norway, sorry.
The historical stuff - castles, museums, churches etc. - is an any decent guide book. If pressed for time, don’t miss the Round Tower (“Rundetaarn”) with its view of all the city spires.
Things not prominent in the guidebooks, yet not to be missed in June in Copenhagen are:
Not worth a big detour, but if you’re anywhere near, check out the submerged statue group in the canal by the Old Fish Market. On a summer day when the water is clear, its semi-abstract grouping of humans and merpeople can just be made out. It’s a weird idea, but it somehow works.
Check out the communal bicycles - you deposit a 20-kroner ($2.50) coin and the bike is yours to use in the inner city for as long as you want. When you park it at any one of the stands, you get your coin back. Nifty concept.
A summer’s day in Copenhagen should involve sipping a big draft beer in Nyhavn, doing nothing but people-watching.
Eat a hotdog at one of the small white hot-dog stands. They’re good.
I happen to think that the Little Mermaid is overrated. She’s very little. And a mermaid. Not a concept that merits a trip out of one’s way, IMHO.
If you’re into modern art and have half a day to spare, Louisiana (slightly north of Copenhagen proper) is a beautiful museum with some very, very cool art in amazing surroundings.
Dragør, a little south, is very pretty with tiny streets, ancient fisherman’s houses etc.
Oh, and Copenhagen’s airport is actually an almost tolerable place to be, a trick considered impossible by most.
Practical stuff:
Cabs are expensive, but trains and buses will take you most places.
Tipping is not expected. It’s rarely refused, either, of course, but don’t feel bad about not tipping.
Most, if not all, people speak English.
Stores close early.
Enjoy your trip!