Hi Kristie,
Stockholm is the perfect home base for a month. For hiking and nature walks, there are many open spaces and parks inside and outside the city that can be accessed by bus or Tunnelbana. There are places to rent a kayak or a bike to discover hidden corners all around the city. Just do a little research.
If you are coming mid-June you could go to Skansen for their Midsommar celebration. It's a nice way to see Swedish culture and hear some traditional music. (Just be sure to pack a raincoat or umbrella and wear layers. Mid-June to mid-July is a beautiful transition time in Stockholm. The days are longer and the nights start getting warmer. Of course, because it's in the North the weather can be variable; cool one day, warm the next.)
You could do day trips around and outside the greater Stockholm area (Uppsala, the archipelago Islands, Mariefred, etc.) and you can take a train to Malmo to experience different Swedish cities. Trains can get you to the mountains in Are. Rent a car and drive to the High Coast or go down to Ystad. You could take a large ferry to Visby or Tallin for a few days too. Catch a smaller ferry downtown that goes out to the closer archipelago islands. Rick Steves recommends a few in his books.
Take a snack and a book to a local beach and go for a really cold swim. Also, it's just nice to relax on the rocks around the lakes.
In the summer Swedes love being outdoors and a lot of restaurants and cafes set up outside tables. It's nice to absorb the vibe and people-watch. Swedish coffee is really good too! Just a small note, in some neighborhoods, restaurants close for part or most of the summer so make sure to check their hours.
Finally, it's an easy train ride to Copenhagen from Stockholm (stop in Goteborg for a day or two if you want), once the month is over. Hope this helps! Have a great trip!
Cheers,
Moomin