Zaragoza and Girona are both great, but you'll find plenty in Barcelona to keep you busy for the time you have there. You need to buy tickets in advance for any of these sights you want to see, because their tickets lines are usually horrendously long, and even if they don't sell out before the day of the visit (which they may), they're likely to be sold out as the day wears on:
La Sagrada Familia
Parc Guell
Casa Mila/La Pedrera
Casa Batllo
Picasso Museum
If the Palau de la Musica Catalana still requires a tour to see most of the interior, that ticket should be purchased in advance as well, because the English-language tours often sell out--though I think not typically very far in advance.
The high number of places for which you really, really need to buy tickets ahead of time makes it difficult to be an efficient traveler in Barcelona. The tickets are timed, so you must guess how long you'll be at a sight and how long it will take you to get to the next one. Then there's lunch; when will you want to eat it (Spain eats early and how long will it take you to do so? Life will be easier if you take advantage of having three full days in the city and book one of the tough sights (listed above) as your first activity each day. At least those you can be sure you will be on time for without having to rush to finish seeing the previous attraction (for which you will have paid a pretty penny; many Barcelona sights are rather pricey). The first time slots are popular with folks hoping to be in front of the rest of the crowd, which isn't a bad idea. Some of the sights are open into the evening, and I'd guess they might be marginally less busy after 6 PM. I'm just guessing, though.
Rome has its own challenges. Both the Vatican Museums (which include the Sistine Chapel) and the Colosseum have recently changed or will change their ticketing procedures before the time of your visit. In the recent past those have been difficult tickets, but for different reasons. There were scalpers grabbing tickets to the Colosseum, and there's some hope that situation will improve. Still, I'd advise you to figure out when you want to go inside the Colosseum and get tickets when they go on sale, if the Colosseum is on your list; the exterior is viewable without a ticket.
he issue with the Vatican Museums seems to be overwhelming demand. They're addressing that by opening an hour earlier and closing an hour later, I think. At the same time, they have halted (at least as of now) the premium-priced tickets one used to be able to buy for early entry before the Museums open to the public. Those early admissions were the best way to have a non-mosh-pit experience in the most popular parts of the (huge) Museums. Now, if you buy first-time-slot tickets (that's for 8 AM entry), you will be part of a much larger crowd than in the past, but the Museums probably won't be quite as busy then as they'll be at 9 AM or later. The Museums offer some tours at extra cost, or there's the less-expensive audio guide option. There are private companies selling tours to the Vatican Museums at higher cost; as of now they should not be promising to get you inside before the general public, but I don't know that they've all updated their websites. In the past, some of those tours included St. Peter's basilica as well as the Vatican Museums, and in many cases the tour groups were allowed to use a short-cut between the Sistine Chapel and St. Peters, thereby avoiding the long walk around the outside of the building and the much longer security line for entry to the Basilica. That back-door situation hasn't been a 100% guarantee, and I don't think we've had a report from someone who's taken one of the combo tours really recently.
The art-viewing experience at the Borghese Gallery is much better than what you'll have in the most-popular sections of the Vatican Musuems. The Borghese is popular (buy tickets as early as you can), but less crowded.
