I appreciate your saying ".....on this trip." As it indicates an understanding of more trips will occur.
Your journey occurs during a time of hot weather chock full of tourist and these factors will impact the efficiency of the trip. Hence a recommendation to rent an apt and stay in Barc for the duration of the visit. Barc offers a rich menu of opportunities to explore and excellent public transportation.
Some thoughts:
Do your research on sites and then use the web to buy tickets for planned times of entry. Be sure and learn the tactics for bypassing lines as a result of your pre purchase of tickets.
Immediately adapt to local clocks for eating times and entertainment. You will have ample daylight late into the evening to utilise to max your experiences.
Do take in a concert at the palau academy of music.
Do plan several daytips. There are opportunities to take a cable car up a mountain to cooler air, go to wonderful beaches and visit ancient roman ruins.
I really hope Enric reads your post as he provides expert advise on finding ways to fully explore his home city and surrounding area.