4 day itinerary with 9, 14, and 16 yr old boys

I do think this is way too much per day. I mean, it's a good rough draft, but I would highlight the three things on each day that are priorities and use the rest as extras if you have more time and energy. Especially with a 9yo in the mix. If you arrive at noon, you may not get to your lodging until 2. You have beach time on 2 days and then plan to go to the Costa Brava. I would eliminate a Beach day or both (have these as extras). I was in Barcelona this past summer with my three kids, the youngest is 11, and we did spend 1 morning at the beach and it was nice, but it took awhile to get to and from the beach and transition to other things (while there are showers at the beach, a better clean-up was required before moving on). Same for Park Guell. It is up the hill from the city, and requires a bus ride and takes time to walk around and by the time that was done, we went out to lunch, and short version, the day fills quickly. Also Park Guell is on the opposite side of town from the beach, so you are talking a lot of commuting that day. But, a lot of what you mention can be seen just by walking around. We saw a lot of Gaudi architecture, La Sagrada, La Rambla, and the Arc de Triomf in one day, walking, so it can be done.

It's unclear to me whether you're arriving in Barcelona from elsewhere in Europe or it is your landing point after a transatlantic flight. If the latter, I think you are way too ambitious on your first day when some or all of you may be severely jetlagged and sleep-deprived. The usual recommendation is to plan to remain outdoors on Day 1 in Europe. I would especially not pre-pay for anything that day, and you have two costly activities planned.

I really liked the Barri Gotic tour offered by the tourist office, but I think the English tour is in the morning, so that wouldn't work for your arrival day. One of the "free" tours fits your schedule better. Understand that the guides have to give a chunk of money to the company for each person who shows up; I think Rick has said it's something like 3.00 or 3.50 euros. If the tour is decent, tip generously. I remember reading positive comments here about Runner Bean.

The Gaudi sites have audio guides, so a guided tour is not your only option at those places. They are all very crowded, and I'm not at all sure that you can get a guided tour at all of them. I can't imagine trying to move a tour group through the sardine-can conditions at Casa Mila/La Pedrera, and La Sagrada Familia also gets quite crowded. If you just want to see the exteriors of the buildings, that's a different story (in which case you should investigate the tourist office's Modernism Tour), but I gather that is not your plan. Going on your own will give you more flexibility in how much time you spend at each place (it was at least 90 minutes at each modernista site for me) and will also be cheaper.

Tickets to the Gaudi sites are timed unless you opt to pay extra for what I call the "wildcard" tickets available for some of them. You must buy tickets to the Gaudi sites in advance to avoid standing in very, very long ticket lines--and sell-outs are possible. I would use the first time slot of the 26th and 28th for one of those that require pre-booking (La Sagrada Familia, Parc Guell, Casa Mila) so you don't have to worry about getting away on time from a previous stop.

Tickets to Parc Guell now include a shuttle bus from one of the Metro stations, so it's no longer a time-consuming commute from the center of town. It's an interesting (if not short) walk back to the center--basically downhill or flat in that direction.

I don't know anything about bouldering at Montserrat. For regular visits one simply goes to the train station beneath Placa d'Espanya and buys a combo ticket that includes both transportation and entry. You'll need to choose which form of transportation you'll use for the final leg of the trip. One of our regular posters doesn't recommend the costlier ticket that includes lunch; I think it is a question of the cost vs. the value.

Prepare your hungry teens for how late they may be eating in Spanish restaurants, many of which do not start serving until 9 p.m. This does not apply to international chains of course.