Europe is a big place. So, you should sort of consider narrowing it down to a particular country. While it would be possible to do a package tour that will take you to several countries on your trip, a much more memorable experience would be to only go to one or two places and spend at least 3 days in each.

In much of Europe, English is the lingua-franca (so to speak - my little joke). Meaning you will be able to get by fine if you speak slowly and don't shout. Also learn to say "hello", "thank you" and "please" in the local language. "Americans" have the reputation for never shutting up and always shouting. I mention this only because it is easy for us to forget when we are rushing to and fro on a trip.

If you want a truly "European" experience spend your 10 days in Paris http://www.paris.org/ or perhaps take a Rhine cruise. http://www.rhinerivercruises.com/rhine_river_cruise_companies.cfm

Or visit the Swiss Alps http://www.myswissalps.com or maybe Bavaria http://www.bayern.by/en/index.html

To economize for your trip you should consider staying in Youth Hostels. In many of them you can get a room for 4 so you would have your own room, and not have to share with others. Join the Hostelling International http://www.hihostels.com (you can do that at your first hostel) and the rates will be much cheaper. The ones I have stayed in have been fine, certainly as nice or nicer than a budget hotel.

If you want to go where they speak English, (more or less)... You could spend 10 days in London and virtually see the world. It is a very cosmopolitan world city and unless you look for it, you might not find much actually English personality to the city any more. (much like New York City or Chicago) But you literally have the world at your finger tips.

The advantage about the UK is that everyone speaks English. The disadvantage is that everyone speaks English.

Where would I fly in and out of?

I have been through Heathrow many times. These days I try to avoid it. It is very congested and they don't seem to be keeping up with the crowds very well. But I do particularly enjoy the airport, many happy memories. Very convenient to London. You can take the express train to Paddington Station (fast, expensive) or the Tube (slow, cheap).

Gatwick is also a good option as is Manchester. Avoid Stansted at all costs.

You might be able to find very cheap flights on Aer Lingus http://www.aerlingus.com/ to Dublin. From there you can get cheap flights to anywhere in Europe.

The airport at Copenhagen is also very nice and not at all crowded. Speaking of Copenhagen, I have also had a very pleasant experience in Stockholm. It is a nice place to be a tourist.

The Frankfort airport is one of the largest in Europe, you can draw your own conclusions on expectations for crowds from that.

How would you get from place to place?
Rarely I have rented a car. Usually I do that when I want to visit some ancient ruin that isn't easily accessible by train or bus. Most of the time I take the train. It is a nice way to see the countryside and meet people. Even when I'm not in a people meeting mood, it is still fun to see how the locals interact. In my (relatively limited experience) Trains in the UK and Europe tend to be much better than what you get used to on Amtrak in the US.

You can get a BritRail http://www.britrail.com/ or EurRail http://www.eurail.com/ pass before you leave often at a discount. If you are intimidated by their websites, don't be shy about calling them. They can help you understand the time tables and such. (Just be prepaired for a bit of a cramp in your phone bill.)

Whatever you do, find out what the locals do, and do that. Ask for recommendations for places to eat and places to visit. Don't be shy about saying you don't want to eat in "tourist" places. Just don't be surprised if you end up eating food that is strange and wonderful. I usually ask police officers or firefighters for recommendations on places to eat. I feel a little more secure about that on the one hand, and usually feel like they aren't going to steer me to a tourist trap on the other.

PS: edited heavily after re-reading the question.

Check student travel agencies

Even if you’re not a student, check student travel agencies. These offer budget fares to non-students as well. Any city with a university probably has such an agency. Try the biggy: STA Travel, Tel. 800-781-4040, www.statravel.com).
Europe Travel 101 - Before you take that first trip to Europe

http://goeurope.about.com/od/europeantripplanning/a/europe_travel.htm

planning trip

http://www.guideforeurope.com/planning_your_trip_for_Europe.html

8 Budget Travel Tips To Make Europe Cheap Again

http://www.bravenewtraveler.com/2007/09/04/8-budget-travel-tips-to-make-europe-cheap-again/

http://www.destination360.com/europe/

Cheap Travel Europe Tour Guide

http://001yourtranslationservice.com/travel-Europe/cheap-travel-europe-tour-guide.html

travelling Europe more guide look here

http://europa.eu/youth/travelling_europe/index_eu_en.html

train coaches and passes is best

http://www.bakpakguide.com/europe/transport/tourstransport.html
idea for the location to visit

http://72.14.235.132/search?q=cache:http://protravelusa.com/EST10D.aspx
http://travel.yahoo.com/trip-view-1102897-10_days_in_europe

http://www.virtualtourist.com/travel/Europe/Switzerland/Kanton_Bern/Interlaken-691572/Things_To_Do-Interlaken-BR-1.html

http://www.travelpod.com/travel-blog-entries/ambough/bellaitalia2007/1189519200/tpod.html

another transport

http://www.autoeurope.com/

A La Carte Europe Tours offer several pan-European suggested tour itineraries lasting 10 days (our minimum duration).

http://www.a-la-carte-europe-tours.com/10-days-tours.html

rail pass

http://www.seat61.com/Railpass.htm#Why explore Europe

First, are there any of those "I really want to see places" for your family? If so, I'd make those my priority and plan around them. In ten days, select maybe three general geographic areas and try to find a central location to stay in each.
The only airport I have flown into or out of is Heathrow. It was fine, and I strongly recommend spending a couple of days in London - there's a ton to see there! Museums, government seats, etc. a-plenty, and I wouldn't miss the Tower of London. One of my favorite sites in England was Stonehenge, if you get away from the urban areas. And Warwick Castle was beautiful, and not too far out of London. In general, museums, castles, and small churches - for Americans it's hard to wrap your head around monuments dating back to 1066. Harrod's has great food, too, which I didn't find out about until the day before we left...it would be a first stop, now.
One dumb tourist thing I like to do where it's available now is take a bus tour in a strange city. It gives me some perspective about how far apart things are, and orients me to their location in relation to one another. It helps to plan effectively, and may rule some site in - or out - that you hadn't considered before.
For travel, I like trains. It's amazing where they go, and they're relatively inexpensive. They're also part of the whole European experience. The train stations are pretty amazing themselves, and traveling via rail (think Chunnel) is a way of seeing additional sites while still getting from point A to point B. The discount airlines have a reputation for being quick, but very uncomfortable.
Have fun, and come back with plenty of questions for the MA community to chew on!

Since I don't know where you will be staying, how far you are willing to travel and what your family likes as far as activities go, I'll give you some great web sites to visit.

One of the best sites I've seen is: EUROPE a never-ending journey
http://www.visiteurope.com/ccm/experience/things_to_do/?lang=en_US

Things to Do in Europe by Category / Things to Do in Europe by Popularity
http://realtravel.com/europe-travel-ideas-d332-2.html

Another is the World Guide, Top Things To See:
http://www.phhotels.com/guides/index.php

and Top 10 Free Things to do in Europe
http://budgettravel.about.com/od/destinationsavings/tp/free_europe.htm