If you have lots of time in Barcelona, consider adding the Sant Pau modernista site if it won't be too much for your children. (This seems like a lot of architecture for kids.) So far, crowding has not been reported there, though they are advertising, and I definitely expect the place to become more popular. It's a multi-building site with very nice grounds. I needed more than the 2 hours I had allowed there, and I think an additional building has been opened since my visit.
One other issue with waiting to buy tickets once in Spain is that buying at the attractions themselves will have you waiting outdoors in what may be miserable weather. Madrid gets seriously hot in summer. Barcelona's temperatures tend to be more moderate, but the city can be quite humid. You will probably have the option of buying online (if there are still tickets available), but I don't know whether you'll need to prevail upon your hotel to print out all those tickets or will be able to get e-tickets. As of 2015 the tourist office beneath Placa Catalunya sold tickets to most if not all the sights I've mentioned. There was a 1-euro-per-ticket service charge, which was much better than having to make a special trip to each place and stand in line. However, the sights may be sold out by the time you get there. You can take a look at ticket availabilty for the immediate future online. How does it look for May 9 and May 10? Keep checking so you can react if the situation looks risky.
Parc Guell is switching over to a requirement that all tickets be purchased in advance; none will be available at the park. I am not sure whether that change will have gone into effect by the time of your trip. Check the official website.
I don't think sell-outs to Montserrat are possible since it's a local train without assigned seats. However, I don't know whether a stroller will be practical there. I haven't been to Montserrat.
I'm pretty sure the Sitges trip is on (or at least can be on) a local train, so that can bena last-minute purchase.
Although you can get to Figueres on the slower regional trains, you may decide you prefer to take the speedy AVE. AVE ticket prices can be rather high at the last minute. If you buy tickets shortly after they go on sale, they are a lot cheaper, but that means committing to a specific date and time. One other wrinkle is that the fast trains go to a station that's farther from the Dali Theatre and Museum, though there is bus service.
People sometimes recommend advance purchase of tickets to the Prado and the Palacio Real in Madrid. Neither of those was a problem on the days I visited, but I can certainly believe there are busy times when the ticket line would be more than a few minutes. It's always smart to avoid free-visit periods at the big-name places, because large numbers of locals are likely to show up then.
Fares from Madrid to Toledo do not vary, but the trains have assigned seats, so they can sell out. Rick particularly warns about the risk of sell-outs on late trains back to Madrid. I love Toledo; I spent 3 nights there and would be happy to return. But parts of the large historic area are seriously hilly. The thought of a stroller in Toledo is, for me, not a happy one.
Buying tickets from the Renfe website can be frustrating. But buying them at Atocha Station can be time-consuming--though the Toledo tickets aren't sold in the same office where I had a 3-hour wait last month. The ticket-vending machines are easy to use if they don't choke on your US credit card.
