What are interesting things to do and places to visit in and around barcelona preferbly some things low cost?

theres loads to do
If you want to go to the beach I reccomend going out of the town, as the city beaches are full of people and tourists in summer, it is cheap and easy to get the train along the coast to castell de fels or sitges, which both have stations within 15/20 minutes walk of the beach.
there are the typical tourist attractions; parc guell, sagrada familia, the olimpic stadium, art galleries if thats what youre interested in
las ramblas street is worth a visit, but expensive to eat and drink on, on either side of the ramblas are the barri gotic and raval areas, which have nice and cheap bars and cafes, its worth getting further away from the touristy areas for cheaper places.
Nightlife is always good, there are hundreds of bars and clubs throughout the city

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1. Walk down Les Rambles and cut into the old city (Barri Gòtic). Visit the cathedral and surrounding areas.

2. Visit Parc Güell and take in all the Gaudí monuments in the park.

3. Visit Picasso Museum and spend at least three hours there. Do walk around the neighborhood around the museum.

4. Walk up and down Passeig de Gràcia - specially around 7pm when everyone is out for a stroll. Just take in the wonderful architecture and do a bit of window shopping.

5. Visit Casa Milà (also known as "La Pedrera" ), Casa Batlló and Sagrada Familia . These are Gaudí's masterpieces. They are incredible buildings. Do climb the towers of Sagrada Familia.