I agree that Norway has spectacular scenery and is a great choice for outdoors activities. It would probably be my first choice.
For a complete contrast to Norway, you might consider looking into either the island of Gotland or the Stockholm archipelago in Sweden.
Gotland is located in the Baltic. You can get there by taking a ferry from Nynäshamn which is about an hour south of Stockholm by train. The ferry takes a little over 3 hours. The main town of Visby still has its medieval walls. I haven't hiked here, but when I visited, I stayed in Visby and rented a bike and rode to several places around the island. There are some nice beaches.
There are many islands in the Stockholm archipelago where you can rent a cabin, go hiking, biking or swimming. Utö is one that I particularly like.
The law that Randy refers to is common to the Nordics. In Sweden it is called "allemansrätt" (every man's right). It basically means that you have the right to enjoy the land and nature, regardless of ownership of the property. There are some limitations and common courtesy applies. If you plan on trekking in the Nordic region, read up on this.
Enjoy Scandinavia. I hope you are planning your visit in July. It really is the best time.