Your choice of cities is almost point to point with great train connections and/or cheap air routes.

Looking like a car for 3 weeks would be far less expensive than 5 rail passes for all these areas - but may consider flying between Italy and London/Brussels area. We do long drives though US (24 hours straight sometimes) so driving doesn't bother us - manual trans is no problem either. Not sure if parking and then transportation within the cities makes the car savings minor or not. Planning to stay in apartments where possible. We may be a little cheap hotel rooms ambitious but the start and end dates are firm - just need to fill in the details - would love any thoughts or suggestions.
Check if you could take a car from the Continent to the UK or vice versa. The cost of the ferry might not make it worthwhile. I recall meeting a Swiss couple (who should be used to high prices) complaining about the cost of taking cheap hotel rooms their motorcycle on the ferry.
I think that your trip is too much. 6 major cities in 21 days means 2 days in each with travel time in between. On top of that you want to go to Cinque Terre (which is out of the way when traveling from Venice to Rome via Florence) and ski for one day in Zermatt. I can't imagine cheap hotel rooms it.
Renting a car for the 3 weeks makes next to no sense. cheap hotel rooms A car is useless in most of those cities. In 3 weeks you just don't have time to drive all that. Why are your only choices a car or expensive rail passes?
Thanks JanisJ - the travel decision was based on cost. The car would be a lease (3 week min) and is much less expensive than renting or trains/flights as far as we have researched. We will keep checking options as we solidify the places and days.
Why fly in and out of Paris, forcing a looped itinerary? Fly open-jaw instead. Yes, too many places for 3 weeks, and a car will be a big liability (that doesn't mean a Rail Pass is a good idea - they are rarely worth the money compared to point-to-point tickets purchased on the national railway cheap hotel rooms systems cheap hotel rooms for each country). You are almost certain cheap hotel rooms to incur huge fines in the ZTLs in Italian cities, and won't be able to use the car there anyway.
One thing that is making your trip more costly and eating up precious vacation time is the RT Paris. That adds the cost for 5 people from Rome back to Paris and eats up a good part of a day. Look at multi-city tickets, flying into London and out of Rome. Fly into London, Euro Star to Paris, fly to Venice, use point to point trains in Italy, fly home from Rome.
If you are already stuck with RT Paris, if you can find a decent priced flight, you could go ahead and fly straight on to Venice. It is a good place to get over jet lag. Use trains around Italy. Fly to London from Rome. Train to Paris.
Point to point train tickets are cheap and trains are great and easy to use in Italy. They take you city center to city center. You, of course, can't have a car in Venice, not allowed in central Florence, and no use in Rome. Paris and London both have excellent subway/metro/tube systems. With the cost of airfare to get you to Europe, cheap hotel rooms it makes sense that you would want to actually see as much as possible, walk and soak up atmosphere, not use days of your expensive trip sitting in a car. Your trip seems concentrated on cities and you will not need, or be able to use the car in most cities. If cities are your base, you can do short day trips with cheap public transport.
Unless you want to see a bit more than Brussels, that is a lot of extra travel time and money spent. I would give that up. If it is set in stone, then add some time, stay in Bruges and visit Brussels from there. That would work better though if you skipped London.
After 2 or 3 days in Venice, train to Florence. Then train to Rome. Fly to London. Train to Paris - finish up in Paris. This will still be VERY rushed but a bit more efficient use of your limited time. I cant see any way you'd have time for the CT -- unless you drop London.
I, too, think your trip is too ambitious. Which places are your absolute priorities? You lose almost a day each time you change locations. I would pick your top 4-5 choices (max) and save the rest for another trip.
London to Brussels will take you about 5 hours, door to door, if you take the Eurotunnel and drive (as long as you avoid rush hour in London, otherwise cheap hotel rooms it takes longer). The Eurostar from St Pancras will be faster.
Paris-London-Brussels-Venice-Rome-Florence-Paris cheap hotel rooms is a LOT of driving with three teenagers in the back of your car. You will not need a car in any of these places, and in Italy, it is easy to get caught out by the restrictions on driving in these towns (leading to expensive tickets).
In fact, the price comparison is not being made accurately. Rail passes almost always waste money. You have to do a price comparison of point-to-point tickets with all the advance-purchase discounts you can get against the cost of the car rental plus the cost of fuel, tolls, and parking.
We have done a lot of point-to-pint train pricing online and again its the x5 that adds up quickly. The car lease is substantially less expensive than the local trains - and I didn't worry about times, I picked the cheapest train fares between cities.
The RT to Paris is set in stone - BY FAR the least expensive option to get across that we found - got tickets for under $800 RT each and booked them. A flight back from Rome to Paris was cheapest direction to fly in/out of Paris - flight from Paris out to Italy was more than twice as much no matter what city we went into.
Yes, Zermatt may be crazy, but its about half way from Brussels cheap hotel rooms to Venice according to driving times and would be a great jewel in the trip - just the view alone would be awesome since daughter is into photography and we all ski. The cost would be high, but we all have those things that we are willing to "make work" and this may be one - we'll see.
Use viamichelin.com to get estimates of drive times and especially fuel and toll costs. In Italy, if you buy your train fares three months in advance, you can get them as low as 9 euros a leg and typically 19 euros a leg. 100 euros for the 5 of you to get from Venice to Florence is cheaper than a rental car sitting in an expensive garage in Venice, then fuel and tolls, then expensive parking again in Florence. Same thing going to Rome. In Switzerland you have to buy a vignette to drive on the highways, 33 euros for that. A car is very likely false economy when you add everything up.
And I'm on the side of everyone who tells you to head immediately to Italy and save Paris for last. No point in staying there twice, save yourself a hotel change and begin your trip in Italy and end in Paris.
We did a 3 week trip to Europe when our kids were the age of yours. Flew in and out of Luxembourg (Icelandic). This was in '84, mind you and I had to use the Thomas Cook train schedule book, Frommer's guides!!.
I think England cheap hotel rooms is your outlier here. We did a different loop than you are (but did do Zermatt--it is glacier skiing, I'm sure you know)--Germany (driving), Switzerland (trains), Italy (trains), south of France to Paris (driving). Train back to Luxembourg.
I would still suggest doing some of the trip by train - the italian part, as you REALLY don't want a car in Venice, Florence or Rome. [we are not just saying that, look at all the threads from people who have been caught by the ZLTs in Florence and elsewhere and you obviously can't use a car in Venice]
my idea would be this - from Paris get the eurostar to London, [book 3 months out to get the best prices] where as I understand it you picking up the car. then drive to Brussels, Zermatt and Venice, where you return it.
We make a list of where we want to go. Get a map and put post -it flags on those place. Gives a very clear indication of what place may need to be knocked off the trip--so far out of the way, really timeconsuming to get to, it frees up a nice amount of time to share over the rest of the trip.
With the age of your kids, it's a great time for you to be going. I think although you think you have a lot time, to do all you want like other posters have said really leaves almost only hours in those cities. Your kids are really of an age that they can go on their own in college, do a semester abroad and even like my son, get a scholarship and spend their junior hear of HS abroad.
Your choice of cities is almost point to point with great train connections and/or cheap air routes. You might also look at www.rome2rio.com which would be excellent for your trip as it prices out car, bus and train travel.jk
Before you decide what is less expensive you have to have all the facts. They are all obtainable with a little cheap hotel rooms research on the web. First, you know you can't drive IN any of the cities you are visiting. They all have restricted zones and huge fines if you make a mistake and accidentally go into an area that is restricted. There are tons of threads on this forum from people who have done that.
Therefore, you need lodging outside of the city centers. Have you looked at how much you'll save/spend on hotels with parking (and then the public transportation to get into the city centers)versus hotels you could walk to from train stations and then walk around the cities.
One of the map sites (maybe viamichelin, might be mappy.com or mapquest) gives you the price of point to point trips including gas and tolls. I have driven in Europe many times and am always surprised at the cost of tolls. Gas of course is about twice what it is in the US.
The main problem I see with your plan is that you are visiting only cities and cities are places you do not want to drive. If you were planning a trip with the main destinations being say the Cotswolds, Provence and Tuscany, then your plan makes sense. But London, Paris, Florence, Venice and Rome?!
As mentioned above, plan your city to city trips three months in advance and get discounted train tickets. I know Italy, France and Belgium all have them but you need to book 90 days out. They sell out fast. So you need to know the dates you want to travel and mark it on you calendar so