We spent our Easter holiday in Barcelona and this was an advise from a colleague. I found Barcelona a friendly city and a good destination for families with small kids.
Barcelona has currently the biggest Aquarium in Europe (L'aquarium De Barcelona), while I knew recently that a new bigger aquarium is being build in Crete.
The aquarium lies on Barcelona beach and you can find there a shopping center (mall) also a teleferic tower (Telefèric de Montjuïc) is very close that takes you the the magic mountain.
The kids enjoyed Barcelona zoo, a family can spend a whole day there and enjoy the dolphin show and there is a great playground in addition to electrical cars and horse riding.
The Zoo is located in the biggest park in Barcelona (Parc de La Ciutadella), in this park you can hire a boat and cruise with the kids in the lake. Also the park has many activities especially on Saturdays where kids can enjoy face drawings etc. The park has also a lot of playgrounds.
You can also enjoy shopping anywhere you go in Barcelona, Barcelona has many shopping centers (malls), in Maria Cristina for instance and also in Catalunya Square. Also throughout the Diagonal Av. Street there are huge number of shops.
Finally there is a tourism bus network where you can buy a daily ticket, this bus has a second floor open space, this bus goes all around Barcelona
Some useful links:
How to use public transportation from/to guide
Barcelona tourist guide