If you're into art, you have three great choices in Madrid, depending on the period you're most interested in. If art isn't your thing, there's the Palacio Real. I don't like palaces, but many people do. I would much prefer to skip the palace and take a day-trip to Toledo. In fact, if I didn't care for art museums, I'd stay in Toledo rather than Madrid. The train fares from Madrid to Toledo do not vary, so you can remain flexible about that possible side-trip. However, the Toledo trains do require reserved seats, and they can sell out. Rick warns in his guide about the importance of buying your return ticket from Toledo in advance so you don't risk getting marooned there. There is also bus service, but the buses are quite a lot slower than the trains.
I'm very fond of both Girona and Cordoba. Time permitting, those would be my day-trip choices, but you may find you need all the days you have allotted to Barcelona and Seville. If you wait until you get to the base cities to decide about side-trips, you are likely to pay more for the Girona and Cordoba tickets, because the promo-fare tickets will probably be sold out by then. Some trains may be sold out as well, particularly between Seville and Cordoba.
I think even your teenagers will be impressed by the Alcazar in Seville. Since you are not going to the Alhambra, the Alcazar will offer your best look at magnificent Moorish architecture. This is a ticket you need to buy ahead of time, because you will otherwise be standing in a very long ticket line. Outdoors. Without any shade. The Cathedral in Seville is another top sight where significant ticket lines are to be expected. You can avoid the line by buying a combo ticket at the Iglesia Colegial del Salvador, which is about 1/4 mile away. Rick explains this ticketing work-around in his guidebook.
The Barcelona sights listed below require timed tickets and have frustrating ticket lines. Buy your tickets in advance to avoid wasting a lot of time. I recommend getting first-time-slot tickets to one of these problem sights each day; that way, you won't have to worry about needing to rush away from another sight to make your ticketed entry time.
La Sagrada Familia
Casa Mila/La Pedrera
Parc Guell (no tickets sold at the park)
Palau de la Musica Catalana (English tours sometimes sell out; ticket lines aren't bad)
Seville and Barcelona both have a lot of very interesting sights. Refer to a good guidebook to see what your options are.