Two adults and four kids (10 - 15 years old) traveling Europe

The key is to find out how astronomical the fee will be for dropping the car off in a different country than where you picked it up.
The rental car companies don't seem to care if you drive it into other countries, (depending on which countries, etc) but they charge a hefty fee for dropping it off in another country.

It’s slightly unclear from you’ve written, but it sounds like you’re planning to rent a car on leaving London to take on your trip around Western Europe. Don’t do this - the driver’s wheel will be on the wrong side for most of your driving.

There are no "must do's". Other than the places and activities that are of importance to your family. So a good place to start would be a family meeting where everyone can say what they most want to see. Go from fhere. Just don't try to squeeze in too many places in that 10 weeks. Trying to see all of Western Europe in 10 weeks would quickly make for a miserable experience.

Is this your first visit to Europe? Reading Rick's Europe through the Back Door might be a good start. Have a look through the Explore Europe section of this website to see what piques your interest.

As for transportation, consider the wisdom of a lease carefully. While it might seem less expensive, you would need a good size van for 6 people and their luggage. Gas is expensive. Parking is expensive. And often hard to find in cities. In fact, driving in most European cities is usually to be avoided whenever possible. Autos are great for rural visits, where public transportation is inconvenient or non existent. But travel between cities by train or bus, or by local transportation within cities, is often the better choice. Advance purchase of tickets (high speed or long distance trains) often comes with a discounted price. Read through the Transportation section under Travel Tips on this site. For an excellent primer on European train travel, explore the Man in Seat 61 website.

Your brood are old enough to be computer-literate. They can raise questions and research issues. For instance, studying how to navigate big cities on public transport (you surely do not want a car in London, Paris, etc.) Then, together, you can piece together schedules. It will come to feel like "their" vacation too.
One thing to think about is back-up, in case someone gets separated. The younger ones might pair off. They all could carry business cards from your hotels and a little bit of cash. On public transport, if someone is left behind at a subway station, the rest get off at the next station to wait for the stray on the next train. (By the way, if they do not come from a big city, the kids probably will enjoy a subway ride as much as going to Disneyland.) And they can take on errands, such as going on their own to buy items at a nearby convenience store.

You'll want a good mix--tourist sites, churches, and museums are all well and good, but you also want to do normal things kids enjoy--maybe a fun park or zoo or something. You'll want some nature--a few days hiking or camping or seeing some natural wonder. You will want to look into festivals and events. Maybe something social as well. You could contact international schools to see if they offer any holiday classes or events or camps at which kids will speak English and where your kids can speak to someone outside of their families. Or other things that connect with locals.

Must do depends heavily on interests. I would consider catching a stage of the Tour de France a must-do, but not everyone would!

I have no helpful information on cars, but be aware of which countries require toll stickers, etc.

Your children should watch some of the videos in the "Watch Read Listen" area of this website to get an idea of what they want to see. Each of the four could rank four sights from 1-4 in priority.

If you're accustomed to renting a vehicle for travel in the US, you may find that renting in Europe poses more potential issues, especially if you haven't dealt with parking and driving in the largest cities in the US.

If you plan to stay large cities like London, Paris, Frankfurt, Amsterdam, Berlin, Munich, Vienna, Rome, etc. a car is a huge liability.
Better to take the train.

If you want to see the countryside and only visit big cities part of the time, consider a mix of rail and rental car. Rent the car after finishing touring the large city and turn it in at the next large city.

If you rent a car, be sure to rent one with the trunk capacity to hold luggage for six persons. That will likely be a larger, more expensive vehicle. Never leave luggage or items inside a parked car unless you want it broken into.

We have driven in Europe including the UK on the left and it has its advantages. However, you need to plan your trip in detail with all lodgings and travel days, etc.

As far as the MUST DO stuff, it depends on where you go. Personally, I would spend at least two weeks in the UK, visiting London, Bath, Winchester, Stonehenge, Oxford, Salisbury and Canterbury. Paris for a week and consider Strasbourg, France, then into either Germany or Switzerland. The Black Forest, Friberg, Triberg or for Switzerland the middle of the country visiting Jungfrau and Lucerne. Italy is a must, do the big three at least, Venice, Florence and Rome, this will take you about 12 days, add the Naples area (Pompeii, Sorrento, Capri and Amalfi Coast) for another 4-5 days.

This already uses up about 7 weeks. Choose between Spain (Barcelona, Madrid, Segovia, Seville, Granada and more) or Belgium, Netherlands and the Rhineland of Germany.

Years ago I looked at leasing a car in Europe. There were a couple of reasonable programs. Google is your friend ! But if your pkans are to visit primarily cities, then I would use trains or planes. Even if you stay in smaller towns, parking could be a problem. Less than 1/4 of the places I have stayed in the past 13 years have parking for guests.

Since you will be there for such an extended length of time you might consider looking at a 2 month global rail pass as it would might be a good buy for you. As other posters have said the car will be a liability in cities. We have done a mix of rail and car rental on our trips. Train is so much easier if it gets you where you want to go, but a car is great for getting to out of the way places.

Here's my thoughts on must do's with kids - at least stops my son enjoyed the most.

Mont St. Michel in France

The Rhein and Moselle in Germany to explore castles and such. Also Rothenburg ob der tuber - be sure to spend the night and do the night watchman's tour.

The Lauterbrunnen Valley in Switzerland. We enjoy staying a week or more. Lots of great hiking and activities for kids.

Venice. We found Rome a bit overwhelming with a kiddo. I was constantly worried that he would get run over by a scooter as they tend to ride right next to where pedestrians are. But so many wonderful sites.

Something else you might want to consider is doing a bike tour. We've only done two. Around lake Constance and along the Moselle. Lake Constance is much better suited for kids.

We are a family of 4 who have rented cars several times. You will need a minivan, and even then it will be tight. You’ll need to pack light. Yes, even for a 10 week trip. Using all seats in the van, you will have very little cargo room. They are smaller than vans in the US. Renting in one country and dropping off in another can be crazy expensive.

As you begin to identify places you're pretty sure will make your final itinerary, keep weather and the calendar in mind. A 10-week trip starting in md-April will end in late June, by which time it is likely to be unpleasantly hot in southern Spain and southern Italy (as well as some other areas you're probably less likely to get to this time around).

If you have a burning desire to see the classic destinations in Andalucia (Seville, Cordoba, Granada) or the heavy hitters in Southern Italy, I'd recommend starting the trip in the hot destinations rather than in the UK, which is sometimes chilly and damp in April. Although it's tough to design a really long trip that hits all destinations at the very best possible time of year, weather-wise, there's no reason--aside from things like family weddings--to plan a long trip that risks a string of 90-to-105F days.

As others have said, your family's interests should heavily influence your itinerary. There are art museums in nearly all major cities, but the car museums I'm aware of are in Germany and Italy. You might want to visit countries where a language is spoken that one or more of the kids has studied, or countries linked to your ancestry.

Travel is exciting. Travel is fun. Travel is educational. But travel is also tiring. The pace your family could maintain over a 10-day trip would not be sustainable for 10 weeks--probably not even for 3 weeks. Be sure to program in stops where you have the flexibility to relax a bit from time to time, like a spot on a lake where you can hike or bike if you want to, or just veg out if you've been pushing a bit too hard.