Italy is my #1 bucket list item (between my interest in Roman & Greek cultures and my parents talking about their time while on TDY in Italy). We would have been there on this trip but they have a big holiday when we would have been there that shut so much down. We decided to take a future trip there. Germany is #1 for my wife, as she lived there in her tween years, and thus the place she mostly fondly remembers from her life as a military brat. We may do Germany with our kids.
Our most recent trip to London was with our nine year old, so not only did we see the cursed child, we went out to the HP studio. It has pretty much all of the set pieces and several full sets. It was pretty amazing. If you do that, I recommend getting in (and out) early. The cursed child was great, but Hamilton is flat out amazing.
If you're interested in going to Belgium also I'll be more helpful. Paris I saw that people already mentioned famous art museums so I'll just add some other museums i enjoyed: Rodin museum- my girlfriend insisted that we would go there but i enjoyed it. Masion de Victor Hugo- one of my favorite authors. Les Invalides- beautiful complex of buildings with several museums and Napoleon tomb(if youre interested). Depends on the date of the trip but London and Paris are great city's to watch some soccer games if you're in too it.
If you can, you should stay over a couple of days in Amsterdam (canals! coffee shops!). As for things to do, both cities are stuffed with incredible museums. Go to the top of the Eiffel Tower, if you haven't before. The view is stunning, day or night. Go to Westminster. Some amazingly famous people are buried there. I wish we were going. We love both cities. You could spend a month in either one and never get bored.
It depends on what your interests are. I was just in Paris and several days outside of Paris in the Loire valley. If they are food I can share some of my favorite restaurants. I will list a possible day trip if you are into wine. Take the tour with the picnic lunch. http://www.chateaudptwines.com The man's winery which is in his family since it was given to his ancestor by Richelieu. Depending on how much you are interested in history you can ask him about it. He'll give you a brief history. The place is beautiful. Sebastian the owner is incredibly nice. His wife will make you a delicious lunch which you will eat in the chateau overlooking picturesque scenery. The food is good and so is the wine. It is also incredibly inexpensive. A tasting is included. It was a definite highlight of our trip. If you don't like wine or have a very strong interest in French history the trip would still be great but maybe not the only day trip you would want to do. If you are only going to see one chateau I would do Chenonceau. It is amazingly preserved and not quite as over the top as Versailles. Versailles is kind of a highlight, but not my favorite part of French history, style, or art etc. I still love it, but it isn't my favorite. Others have talked about the Louvre and the time required. It is okay to just hit what you want to see but remember you will go back. I enjoyed the Catacombs a lot as well. If food is your thing, there is are some really cool cooking classes that include going to the market and purchasing the ingredients with a chef. London is a blast. There is so much to do that it depends on your specific interest. I would say that taking a tour of the Tower of London and the museum is the one must-do.
London is my least favorite city I have visited in Europe. Don’t take the black taxis because they are a rip off. They also have Uber there which is helpful. Paris, the Eiffel Tower is a must, especially at night. Take the hop on hop off bus tours if you’re short on time.
Got back from Paris Friday. The Catacombs are closed due to a strike, so do that next trip. Everyone's recommendations above are solid, although I would ads the Sainte Chapelle (a block from Notre Dame) which is the most beautiful church I have ever been to. The Louvre is open nights on Wednesday nights, so either go then or Thursday morning (get there as close to 9am as possible if you can, it gets chaotic post 11am). http://www.lesediteurs.fr is really yummy.
The V & A Museum in London is great. The tanks in the Tate Modern are kind of spooky. There is an old secret Freemason's lodge at the Hyatt Andaz Liverpool station that is fun to visit. You need a tour from the hotel front desk.
I went to school in London in 1993. Went to Paris during Spring Break. This is an easy one. You have to go to the British Imperial War Museum. I'm not typically a museum guy, but I loved every part of it. I'd also recommend going to an old brewery. Their history is a lot longer than ours. And if you can find a Kiwi, have them take you to the Church. They'll know what it means. As to Paris, the one off-beat thing I really enjoyed was going to the Pompidou Center. Yes, Eiffel Tower and Notre Dame are nice. The Louvre I hated, but the Pompidou center was just an interesting thing to go in and explore. Oh, and going to see Jim Morrison's grave is also interesting.
I have been to London times and Paris twice. In each city I would take a Big Red Bus tour. The busses are good for oriented you with the city and the tourist sites. Generally you can hop on/off at each stop for the day of your ticket. Some of the tourist sites are not much more than photo ops, so the bus tour easily covers these. In London I would do the following: Tower of London. They have amazing guided tours every 30 minutes or so. The tour we had was given by an ex-RAF veteran. Simply one of the best public speakers I have ever heard. See a play or two or three. London and NYC have the world's best theater scenes. Get a TimeOut London magazine. The British Museum. See what the British Empire brought home with them. Drink a British beer in a pub. Order some fish 'n chips for lunch, while you are there. Have dinner at an Indian restaurant. Have a afternoon tea. British cuisine aint great, but afternoon teas are amazing. In Paris, I would do the following: Musee d'Orsay. Less crowded than the Louvre. Impressionists paintings are outstanding. Climb Eiffel Tower and Arc de Triomphe. Elevators at Eiffel Tower are available, but waits are long and cost $$$. The Orangerie Museum. Has a Monet exhibit worth seeing. Avoid the Louvre. Crowded and after a full day there you will leave feeling you have not see that much. If you go in the offseason and have three days to explore ... Stroll down the Champs Elysées (near Arc de Triomphe) Eat dinner where the locals eat. The locals have high standards and you will be impressed. Take a day trip to Giverny, Monet's home. See the Japanese gardens in his paintings. Your hotel will help you book the trip. Take a day trip to Palace of Versailles. See how the top 1% of the top 1% lived.