Scandinavia is not known as budget-friendly location, but there is actually a lot to do that costs nothing or very little.
COPENHAGEN
Free:
Window-shop in Illums Bolighus and wander the Stroget
Ride to the Little Mermaid statue (and surrounding park)
Ny Carlsberg Glyptoteket (free only on Tuesdays)
Find the sidewalk trampolines
Wander the meatpacking district
People-watching in any park or town square
Get into a bike lane and just ride around. Best way to quickly feel like a local.
Not free:
Louisiana Museum of Modern Art with seaside sculpture garden
Go see a pro football (soccer) match
The best cinnamon bun (practically free on Weds)
Torvehallerne food market (with the best tacos)
Cheap stuff at Flying Tiger
City views from the Round Tower or from this church
STOCKHOLM
Free:
Hötorget Square’s daily outdoor market (with the Hötorgshallen food market nearby)
World’s largest art museum (a.k.a., the subway!)
Use the bathroom in the Grand Hotel
City library (especially for architecture buffs)
Free walking tours (tips strongly encouraged)
Not free:
Icebar (book online ahead of time)
Vasa Museum (almost-intact 17th century ship)
Grona Lund (amusement park)
Bike rental (ride around Djurgarden)
Jaime Kurtz, Ph.D. 