Barcelona during Holy Week/Easter

No, you won't have any problem finding restaurants and tourist attractions open. Some will be close but most of them, the more popular at least, won't. Most shops on the other hand will be closed on Friday and Monday (and Sunday, as they always are!). Saturday should be a regular business day.

Expect extra masses at the Cathedral. Other than that, nothing. There is one procession on Good Friday in Barcelona but it's put on by Seville expats and not connected to the church. Barcelona is the least religious city in the least religious part of Spain.

If you were there on Thursday and wanted to see some Semana Santa activities, I'd suggest fast training to Zaragosa for Holy Thursday. One of the most amazing experiences ever for me. Very energetic processions with hundreds of drums go all day and night. There are several routes in town and the processions are put on by about 26 fraternal organizations of varying sizes (think Elks club) for which this is the big event of their year.

While Brad's suggestion on an escapade to Saragossa is indeed a great idea, if processions are your cup of tea I'd like to suggest a couple of possibilities much closer to Barcelona and very spectacular too:

GIRONA: http://infocatalonia.eu/w/4gGMr
TARRAGONA: http://infocatalonia.eu/w/s5hrO

Both are at less than one hour from Barcelona by fast train.

But if you really want to experience a traditional medieval procession Verges (near Girona) should be your choice: http://infocatalonia.eu/w/20KB8 The town of Verges, is famous for its Dansa de la Mort (“Dance of the Death” or “Macabre Dance”) celebrations every Holy Week, probably the last remaining Dance of Death in Europe, performed uninterruptedly since the Middle Ages.

Some, not all places will be open. Certainly smaller family operated establishments will close, or have private parties. Keep an eye on the weather. Most processions will cancel in the event of heavy rain in order to protect the statuary, relics, and costumes.