Hi Siri, regarding travel during Semana Santa, the whole week is a school holiday in Spain, so many families will traveling during this period, also some of the more popular train lines between the major cities will see an uptick in the number of passengers. You should probably secure your tickets beforehand to avoid sold out trains. Note that Easter Monday (April 22) is a public holiday in Spain, so the large majority of shops and restaurants will be closed.
Unfortunately there is nothing really spectacular going on in Barcelona in relation to Holy Week. We just do not have the same intensity of Semana Santa as one would find in the southern part of Spain. If you want to see one of those emotional Semana Santa processions, your closest and better option would be in Cuenca. However since Cuenca is a bit too far, Zaragoza also has some more traditional Semana Santa processions, not on the scale of other parts of Spain, but still fairly decent.
Within the smaller towns of Catalonia, but outside of Barcelona, one can find some interesting holy week celebrations too, here's a quote from a fellow Barcelona contributor, Enric, regarding holy week around Catalonia:
In Tarragona -former Tarraco, the glorious Roman capital of the provinces of Hispania Citerior, and of Hispania Tarraconensis, which covered most of the Iberian peninsula during the Roman Empire- not only you'll find plenty of examples of magnificent architecture of that period, including city walls, forum, amphitheatre, circus, temples and arches, but also a great performances(#) during the Holy Week, such as the parade of the 'armats' or the 'processó del Serrallo'
Similarly, in Girona -a city with a beautiful medieval centre, which was the scenario of several chapters of Game of Thrones- one can find many displays of spirituality such as the 'manaies' -spectacular parades with cohorts of legionaries, reminder of its Roman past, honouring the death and resurrection of Christ.
Worth mentioning the Easter Procession that takes place on Maundy Thursday in the sleepy town of Verges, at a stone throw from Girona. The Procession of Verges is a theatrical representation of the life and crucifixion of Jesus Christ which includes the deeply moving Dance of Death, probably the last remaining Dance of Death in Europe, performed uninterruptedly since the Middle Ages. If planning to attend, note that the town gets a lot of visitors during those days so if you intend to stay for the night procession -which is where the Dance of Death takes place- prebook a.s.a.p. your accommodation. Also, note that while the street procession is free to attend, the theatrical performance requires a ticket which can be purchased here (a few weeks before Easter).
Hope this helps! :)