My wife and I made our first trip to Barcelona back in mid-May this year and enjoyed it immensely. There is just so much to see and do and the city is very vibrant and very clean. Now to answer some of your questions.
1) If you are there on a weekend you must get up to the top of Mt Tibadabo. They have a great little old fashioned amusement park that your son would just love. I know he may be Disney'd out by then, but even a couple of rides atop the mountain along with the slides that I enjoyed along with the views of Barcelona below make the trip up to the top worthwhile.
2) And speaking of the trip up to the top. You can board an old Blue Tram (the only one now running in Barcelona I believe) and take a nice rickedty ride part of the way up and then catch a Funicular to the park itself. You can just walk around if you don't want to go on any rides. There is also a beautiful little church on top which again I recommend going inside for a quick look.
3) Another must see, especially with a 4 year old would be the Magic Fountain of Montjuic. The huge fountain in front of the city's art museum has a number of shows starting a dark on Thursday through Sunday. The shows are a colorful display of lights, water and music.
4) You might want to consider doing a tour of the Barri Gotic (Old Town) district. We did a tour with a company called RunnerBeans which was let by a Barcelona native. All tours are in English and you pay only what you think the tour is worth. They also have a tour for kids specifically. If you check out their website you can learn more about it.
5) As far as food is concerned I think you should try out St Josep La Boqueria the open air market under a roof just off Las Ramblas. If you can't find something you all like at the number of places to buy and purchase food inside I would be very surprised indeed.
6) You should also spend some time on one of the beaches. There is some full nudity on the beaches, but it is very limited. I wouldn't be put off by that. It doesn't seem to bother anybody.
You can check out my pages for additional information and now that I've written you in response I think I will go add a few more pages so make sure you stop back. If you have any other specific questions I'll do my best to answer them based upon our 4 day trip.
Also you can get around very easily and cheaply on the Barcelona Metro. Very clean and very safe.
Fondest memory: Too many to only highlight one.
The highest hill to the north of the city is called Tibidabo. Many of the streets on the way up the hill are lined with some of the most expensive homes in town. Needless to say, the hill offers up some spectacular views over Barcelona and out to sea.
To get up to Tibidabo by public transport, first catch the FCG train to Avinguda del Tibidabo. From here you can catch the gorgeous little Tramvia Blau (blue tram), the city's last surviving tram, which winds its way halfway up the hill. It operates most days, during daylight hours only. At other times it is replaced by a bus. The tram (or bus) will drop you at Placa del Doctor Andreu, and from there you can catch the Tibadabo funicular, through the woods, to the top of the hill (the funicular was closed for renovations when we visited in Jan 2007).
Fondest memory: Once you make it to the top of the hill you will find the Parc d'Attraccions which is an amusement park (refer to the website for opening hours), and a popular place for the locals in the warmer months.
Needing something a little more cultural - head to the Museu de la Ciencia (Science Museum) or the Museu-Monestir de Pedralbes - an old convent which has been converted into a museum of monastic life.
You can't miss the impressive Temple del Sagrat Cor - the Church of the Sacred Heart, which was built based on Paris' Sacre Coeur; or the Torre de Collserola, a 288 metre high telecommunications tower designed by Sir Norman Foster.
When I went up the Tibidabo Hill it was cloudy and raining, so I couldn't see really much of the magnificient view of the city you can admire on clear days.
Anyway, it was not bad, so I took this picture, better than nothing, isn't it?
To get here, take the Blue Tram at John Kennedy Sq. (Avinguda del Tibidabo FG Stop)
"El Tibidabo": There is a small amusement park on the top of a hill, different cool attractions there! But better than the attractions the beautiful views of the city! There is a modernist church as well you should visit.
How to get there: Take a view to the trasportation tip.
Fondest memory: The views, views, views...
As all modern cities around the world Barcelona has a comunications tower too!
Views from there are absolutely great!! Near "Tibidabo", you can visit both places at the same time.
The tower has 288 meters.
More info, schedule and prices: http://www.torredecollserola.com/
At 542m (1778ft), Tibidabo is the highest hill in the
wooded range that forms the backdrop to Barcelona. If
the air's clear, it's a great place for views over
the city. The locals come up here for some thrills at
the amusement park Parc d'Atraccions, which has rides
and a house of horrors. As hair-raising as anything
at the Parc, however, is the glass lift that goes
115m (126yd) up to a visitors' observation area at
Torre de Collserola telecommunications tower. The
more sedate can find solace in Temple del Sagrat Cor,
Barcelona's answer to Paris' Sacré Coeur; it's even
more vilified by aesthetes than its Paris equivalent.
Looming above Tibidabo's funicular station, it is
actually two churches, one on top of the other. The
top one is surmounted by a giant Christ and has a
lift to the roof.
Favorite thing: Take a trip to get this magnificent view of the city. There is a train taking you all the way to the top as well as a bus but the train is more fun.
Favorite thing: This is EL TIBIDABO,a mountain outside of Barcelona,where you can see this beautiful temple and also a fun fair.