Swimming pools and lidos
Swimming pools of all sorts are typically run by swimming clubs (Club de Natacio) and to enter you have to pay a day membership. Barceloneta and Can Drago both have outdoor lido-style pools. Further away there are water parks at Illa Fantasia (Premia de Mar) and Park Aventura.
We were a little disappointed by swimming pools in BCN. It's not that there aren't any - actually there are lots and lots and the pools themselves are all of very high quality - but they are typically part of sports clubs which you have to be a member of to use. Membership is a little like joining a gym and relatively expensive unless you plan to use the facilities at least 3-4 times a week.
If you do want a swimming trip, you can get a day-membership to access the pool, but the cost is quite high - typically around 8 euros just for a casual dip which works out at 40 euros for a family of 5. However, we have now found out that there is quite a range of prices for pools in Barcelona. Whilst the 'big' centres like Barcelonetta are more expensive at €10-15 per entry, Can Filipe in Poble Nou is around €7.30 and €4.20 for children, so do look around.
Can Drago
A better bet is to spend a longer time at somewhere like Can Drago lido - also run like a sports club, but a more pleasant place to spend an afternoon. This is along Meridiana to the North East of the city and can be reached easily from metro line 2 (Puig y Fabra) or line 4 (Llucmajor). If you go mid-week the rates are lower, although the outdoor pools only open during the summer months. And be careful about Sundays - often places close on Sunday afternoon.
Alternatives include Barceloneta Swimming Club which has outdoor and indoor pools and is right on the beach. Next door is Barcelona Swimming Club with similar facilities. Both are reknown for their water polo and you may be able to get tickets to a European cup match once in a while. Barceloneta are currently Spanish champions.
Aqua park - Illa Fantasia
If you are feeling adventurous, Illa Fantasia is a large Aquatic Park with lots of pools and slides a short train ride away along the Maresme - it's about 17 euros for an adult, 12 for a child, but worth it for the range of rides and water features and a really good place to spend a day with water slides, wave pools and everything. Take the train to Premia de Mar, then catch the bus up to the Aqua Parc (or a 30 minute walk from the station if you like to stretch your legs). You can get all-in-tickets from the Renfe office or the rodalies ticket machines.
Alternatively it's at the junction of Vilassar de Dalt and Premia de Mar off the autoroute if you are driving.
Further away is Port Aventura just to the south of Tarragona - run by Universal Studios this is a rival to Disneyland and has both a water park and amusement park. We've not visited but tickets can be bought directly from Barcelona stations and it has it's own train station just before Salou.