I would love to take my 18-year-old, boy/girl twins to Europe this summer before they start college. I want them to have the opportunity to explore on their own as well as time for family adventure. We are all very active. One is outgoing and loves the sun and one is much more shy and does not like hot weather so it can be hard finding a balance. They would both love to visit/stay in a castle, hike, and explore historic countryside as well as visit museums. What would you recommend for a good, first-time, two week vacation to Europe with these two?
Fly to London. Spend a couple days there. Do a short trip out from London to someplace historic and interesting (plenty to choose from). Back to London, Eurostar to Paris. After a few days in Paris, grab a car, head out to Normandy, Mont St. Michel. Back to Paris, fly home. If you must go in "summer" go in June, as early as possible, before the worst of the heat and crush of crowds hit.
You might let each of them plan part of the trip. Say you all sit down and think about possible destinations. Maybe let each of them pick a destination and do the research on where to stay, what to see, what activities to engage in, etc. Or maybe you pick all the destinations, but give each of them one day at each location to plan the activities for that day.
Eighteen is certainly an age when they should be capable of planning part of this trip. And letting them do that will get them more invested in it.
You could even ask each of them to plan a day they think their sibling would enjoy.
Your family's interests suggest that a 3 centre holiday will optimise your time doing things that you want to do, but minimising lost travel time between destinations. Fly open jaw.
Exactly when would you be travelling? Early June is less busy than August.
Get each family member to come up with their own itinerary and see if part of one can be combined with another.
London is a good place to start for your museum fix - many are free. From there, you could head to Wales or Scotland for your walking and castle time. From there, head south for some sunshine for a few days of chilling before you return home. Portugal (Algarve) would be less hot than say Italy or southern Spain.
Be aware that in most European countries 18 is old enough to buy alcohol either in shops or bars. Make sure they don't get over-excited by the possibility.
Lane made a good point about getting them involved with the planning. With two teens who have diverse interest, neither is going to get to go to / see / do everything he / she wants. And each of them is going to have to go to / see / do something they're not crazy about. But the more involved they are the less likely you are to have to deal with "unhappy campers". Have you ruled out taking one of the tours offered by Rick Steves? Might save a lot of headaches if taking one alleviates the differences of opinions and is financially doable.
I'd suggest the UK, London and points north. Plenty of castles and museums. Fly home from Edinburgh or Glasgow or Manchester or some other city.
Alternatively, have either or both of them studied a foreign language in school? Liked it? If so, consider a country or countries where they can hear and use their language(s). Like Paris and beyond in France (Dordogne has many castles, and in Guedelon they're building one the old-fashioned way.) Or Spain, Germany, Italy.
Let them decide as much as possible about the trip, and enjoy their prowess in planning and traveling.