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).