OK, well, maybe I should re-post this as "Great things to do in Barcelona for less than a Euro" as this is not strictly a free activity, but for 70 cents, you can't complain...
Montjüic mountain in Barcelona offers a great escape for families and anyone really wanting to escape the hustle and bustle of Barcelona. There are some fantastic views of the city from up there, as well as the Olympic Diving boards, Botanical Gardens, A Cactus park, the Greek theatre and the Miró foundation, and of course the cable car ride up to the top of Montjüic castle.
Barcelona's tourist bus stops at various places in and around Montjuic, but as I'm trying to offer some money savers for the coming months, then I can highly reccomend taking the Funicular de Montjüic.
The Funicular is a bit like a train, but it's pulled on a cable and while one goes up, the other goes down. Funiculars are used to climb inclines and can be found at the back of Barcelona at Tibidabo and also on day trips outside Barcelona at places like Montserrat.
It's a fun way to get half way up to Montjüic and if you can get a seat at the front, you'll see the oncoming train as you go up/down - the trains are without drivers, and controlled solely by the cable.
The Montjüic funicular leaves from Paral·lel metro station, on the entrance next to Carrer Nou de la Rambla. It's included in the metroploitan transport area, so it can be used as one trip if you arrive from the station from elsewhere (you don't need to pay again, just walk straight through, as if changing lines on the metro) and if you arrive there on foot, you can just use 1 trip on a T-10 journey card which is the equivalent of 70 cents - so come on guys, almost FREE?!
Don't forget, you can also make considerable savings on accomodation choices in Barcelona with cheap apartments to rent in Barcelona instead of going for the more traditional hotel rooms, for example.