Scandinavia is great in the summer – you’ll like it! If you go from mid-August onwards school has started in Denmark and there will be fewer people around (e.g. in the Tivoli Gardens, that is a must visit!).
There are so many other sights in Copenhagen too, and it is very easy to walk around. You need to go up the church tower in Vor Frelsers Kirke, you climb on the outside (!) of the increasingly narrow tower and have a great view (can be scary, but there is a bigger platform further down).
In the same area there is Christiania; a “free-state” founded in the beginning of the 70s by “alternative” people who simply moved into deserted military barracks and created their own society within the society. They had a well functioning society there for a long time, however they didn’t pay tax or rent, so after long negotiations they now do. Traditionally they were VERY ok with light drugs and had people coming from all over Sweden & Denmark to get it there. You can take a guided trip around Christiania led by people living there in English. It is usually very popular.
You can go north on a train and visit Louisiana, a world-famous art museum in beautiful surroundings.
Nyhavn is where people go drinking in summer, around an old canal. Have a look round there, very quaint and great in summer for beer drinking… And do go on a canal ride!
There is a great science park-type museum called Experimentarium, good for kids.
You also need to go on the DLR-like metro, without drivers, to explore a recently developed area of Copenhagen – interesting architecture and good for walks.
There are quite a few good arts museums too.
Obviously, you need to bring your swimwear; there are places to go swimming almost in the middle of the city!
It is easy to drive in Scandinavia; fewer cars on the roads and generally wider roads than in the UK (not counting motorways). You could drive into Sweden (crossing Europe's longest bridge, which is, in fact a) a 4km tunnel b) a 4km artificial Island and c) an 8 km bridge. The South of Sweden is really nice in summer - lots of long beaches, quite deserted, so if you want solitude you'll find it, or if you want to people-watch, there is scope for that too.
The South of Denmark is nice too - you could easily combine them,
I'm not sure if you have any children, and/or any special interests, but I could pass on a few suggestions if you write a little more about what you have in mind.