We did Paris and London this past March, but I can’t help with Barcelona...
we stayed at AirBNB’s in both cities because we are a family of 5 and hotel rooms are impossibly small. We needed space. My children are a bit younger than most of yours - 17, 15 and 12 - but they all loved the trip. This was our itinerary:
Day 1 London:
- Hop on Hop off bus tour - Big Bus Tours (There are several companies that do the same thing. I think I would go with the Original Tour if I was to do this again. The buses looked more modern and seemed more frequent.) It allowed us to get the "lay of the land" and drive by some of the things we weren't planning to stop to visit. When we went the weather was horrible and cold. All of these buses are open on the top which made the entire bus COLD. I'm not sure it was worth the expense overall and I would skip it next time. We also did a tour of Westminster Abbey, which was really impressive.
Day 2: Toured Tower Bridge and the Tower of London. We had a great time doing both. The Tower of London houses the Crown Jewels. The ravens there are so cool.
Day 3: Day trip to Stonehenge with Evan Evans Tours. Worthwhile and fun. We just did the Stonehenge Express Tour since we were limited on time, but they also have day long tours that include Windsor, Bath and Oxford. Also, went to the Churchill War Rooms when we got back to the city. My oldest two really enjoyed this. It was fascinating.
Day 4: Went to Greenwich and toured the Royal Observatory and stood on the Prime Meridian. This was fun! The town of Greenwich is so cute and there is an amazing view of London from the hilltop in front of the Observatory.
After getting back into London, we went to the top of the Shard for some great views. We went right before sunset, so we saw the city in the daylight and as it lit up into the evening.
Day 5: Changing of the Guard at Buckingham Palace (get there early! It gets very crowded), the Borough Market for lunch (a really neat experience) and to Piccadilly Circus for the evening to shop and look around.
Day 6: We took the Eurostar from St. Pancras to Gare du Nord. The trip was about 2 hours and very uneventful. I was expecting some kind of announcement that we were beginning to go through the Channel Tunnel, but there was nothing. Upon arrival at the station in Paris, there are many people offering unauthorized taxi rides, but the official taxi stand - with licensed taxis - is to the right when you get to the front of the trains. You will get in line and there are uniformed men directing you into a taxi. Once we got to our apartment, we walked to the Eiffel Tower and went to the very top. It is obviously a must do.
Day 7: Toured Notre Dame and Saint Chapelle, walked along the Seine. All beautiful!
Day 8: Arc de Triomphe, Champs Elysees and the Tuilleries Gardens.
Day 9: The Louvre. It is overwhelmingly large and at times frustrating. I knew my kids would not last for hours on end, so I picked a handful of things to see and even that took us the afternoon.
I spent a lot of time picking and choosing what to do because we had very limited time in each city, and since we were traveling with teens, I had to keep them in mind. We missed a lot because of those factors, but overall, we got a nice taste of each city. We can always dig a little deeper on future trips.
Random things of note:
Big Ben is undergoing a massive refurbishment, so it's not visible, it's not chiming and there are no tours. Parliament is also under refurbishment, but it's on a much smaller scale.
The London Eye ferris wheel is exorbitantly expensive.
Most places offer discounted student tickets with a student ID.
All of the museums in Paris are closed on Tuesdays.
Traffic is nerve-wracking in both cities. We did not rent a car and relied on public transportation, Uber or walking for the entire trip. It worked out great.
The Sky Garden in London is free to go up and the views are similar to the Shard.
Hope that helps some. If I can be of any help, feel free to PM me. Have a great trip!