Barcelona
 

Does anyone know any free activities such as museums in Barcelona?

Parc Guell is free.
Hospital de Sant Pau (a UNESCO heritage site) is free to walk around the grounds.

I don't know about free museums but you can certainly walk around the city and view all the wild Gaudi architecture.

In addition to Parc Guell and all the Gaudi buildings you can see from the outside, La Boqueria market is free to walk around in as well. It's not a museum but a worthwhile site in its own right.

Walking around Las Ramblas is free also and there's plenty to see there, although the various street performers do accept "donations."

In addition to the suggestion already posted: Parc de la Ciutadella, the beach, Santa Maria del Mar and Montjuic come to mind right away.

The Museu Nacional d'Art de Catalunya is free on the last Sunday of every month.

In the late summer each neighborhood has a festival (fiesta major). These are free. And if you can find when and where they are performing or practicing, Castellers shows are free.

Watching the Catalans dance the Sardana on Saturday nights and Sunday mornings in front of the Cathedral is free.

The view from Montjuich and Tibidabo is the cost of underground.

The Magic Fountain in the evening was free.