Museu Picasso is quite a decent museum, with a good collection of Picassos and other artists (there's the main collection plus a temporary exhibition). It's located in Carrer Montcada, a street where the richest lived - the building is an old palace.
The museum is expensive though - 9€! But it's free on Sundays from 3 pm and free the first Sunday of each month, for everybody.
Oh and when you finish, have a delicious ice cream round the corner, at Passeig del Born, while you contemplate the gothic church of Santa Maria del Mar (in my opinion, it beats the cathedral). 

Small supermarket which caters for expats and lovers of british food. You can find curry pastes, marmite, british cheese, treacle, crumpets, etc.

Renoir Floridablanca belongs to a small chain of cinemas which show original version films (not so common in Spain). Specialised in not-so-commercial films, mostly European or independent films.