Ireland, Scotland & England trip

Considering that we spent 2 weeks in just the South of Ireland, I would say that 3 weeks for all 3 countries is a lot to try and fit in.
What do you like to do? What are sites that are on your list?

I would pick the one that interests you the most. All 3 in 3 weeks sounds completely crazy to me.

I haven't been to Ireland, and spent only a few days in Edinburgh, but I've been to England several times. I suggest flying into either Shannon or Dublin, renting a car in Ireland (except for Dublin), then flying from Dublin to Glasgow or Edinburgh, renting a car there, seeing what you can or want to in Scotland, then driving into England, dropping the car near London and spending your last days there before flying home. Or all the above in reverse. With three weeks you'd be moving around a lot and obviously could just scratch the surface of those three destinations. (Ireland is a country. Northern Ireland, Scotland, and England are parts of the United Kingdom, at least for now.)

I suggest driving -- except in big cities -- because it will give you flexibility and let you explore rural areas and smaller towns, and because there are five of you and trains will get expensive. But driving on the left side is a bigger challenge for some of us than for others.

I also think you'll have a better trip if you don't try to see all three areas in three weeks. I'd suggest England and Scotland, flying "open jaw" into one and home from the other. Look at the "Explore Europe" section of this website and the RS Great Britain guide to see what sights interest you most, plot them on a map, and use a tool like rome2rio and/or Via Michelin to develop routes and travel time estimates.

Keep in mind that moving around costs time and money, sometimes you have a better trip with just a few destinations and more time in each. And that the group will move as fast as its slowest member, which I'm guessing will be your youngest child.

That's a very tough question to answer - there are literally hundreds of things to see and do. So, my best piece of advice is to get yourself some guidebooks (Rick Steves has some fantastic ones :-)) and do some reading!!!
Come up with a list of what you want to see and see where that leads you - and then zero in on how you want to divide your time.
Things to keep in mind: each change of location is going to require anywhere from 1/2 to 3/4 of a day.
You won't be able to see/do everything you want - set priorities and involve everyone!
Plan with the thought that you will be back!!

Whether to rent a car or take the train will depend to a large extent on what you want to see. For example, if you want to see the Cotswolds and Lake District in England a car would be your better choice. If you are planning to visit biggish cities like London, Edinburgh, Glasgow, Oxford and Cambridge then a train would be the better choice

We have visited Ireland, Scotland, Wales and Britain in the past few years, as well as London some years ago.

We did a wonderful Celebrity 14 day cruise of the British Isles and stayed three days in Southern England renting a car and visiting Salisbury, Avebury, Stonehenge and Windsor Castle.

Both my Wife and I find that the vast majority of our ancestors came from Britain and Ireland and we planned to visit some of the places where our ancestors came from as well as key places. We decided to do a 4 week driving tour and decided not to visit Scotland and Ireland on this trip. On our cruise in 2013, we visited more ports in Scotland and Ireland than England. Still, we will likely do another trip for Ireland and Scotland at some time in the future.

The more research that I do, the more things that I want to see. Our driving plan is to avoid driving large distances on a single day. We have it mapped out and only have two days were we are driving more than 2 hours.

One thing that we learned driving in England in 2013 was that driving through the beautiful English countryside takes a lot more time than in North America.

You can see all three countries, but I suggest that you eliminate Ireland, at least. Do it later.

Also, don't forget Wales. We visited north Wales when we visited Liverpool in 2013. It was very scenic. Also, there are many castles in Wales.

If you plan on spending a lot of time in the large cities, London, York, Edinburgh, etc. don't rent a car. I advise you to try to do some touring by car, the countryside is amazing. We plan to visit only England and Wales and will visit Balh, Cardiff, Tenby, (more of south Wales), the Cotswolds (Stratford on Avon, Oxford, Blenheim Palace, Warwick), York, Whitby, Durham, Hadrian's Wall, the Lake District, Winchester ending in Southampton.

There is a great book that can assist on road trips in Britain. Eyewitness Travel, Back Roads of Great Britain.

If you want to see England and Scotland on this trip, I suggest spending a few days in London, then renting a car head to Bath, the Cotswolds (you could continue to Wales, but that would take 3-4 days), then on up to York, which is worth 3-4 days, then on to Durham, Hadrian's Wall, on to Scotland (unless you want to spend a few days in the Lake District in NW England). Scotland should start with Edinburgh, continue north to St. Andrews, as well as other sites in that area. You can continue north toward Lock Ness and Inverness with other places on the way, then continue down the west of Scotland toward Glasgow. Check out flying into London and out of Scotland.

If you just want to get a 'taste' of each, then yes, you can do that. NO WAY can you fully see all three in three weeks; you couldn't see ALL of any one in three weeks- though you could see a great deal. If you really want to expose your kids to all 3 countries on this one trip, then I'd suggest something like the following:
If you are willing to drive, Ireland is the place it'd benefit you. We didn't (and still don't) want to drive in the UK or Europe. We did a package tour to show our kids a small part of Ireland. If you go that route, pick one that is for 5-7 days. If you want to drive, you'll need to ask for more help on this site. Then fly to Inverness. Stay for 2-4 days and do some day trips (there are LOTS of good ones and Inverness has THE best Tourist Office). Take train to Edinburgh. Stay 1-2 nights. Take train to York. Stay 1-2 nights. Take train to London. Stay there the remaining time. Depending on how you allocate your days you will have 5+days. Plan on one to three day trips by train from London. If you want more specific suggestions, let me know. We did a similar trip when our kids were 9 and 17.

The guy who sponsors this Forum has some great itinerary ideas.

Ireland

Great Britain

England

His pace is a little rushed for us, but you can cobble it together to suit yourselves. 3 weeks will fly! I would encourage you to spend at least 4-5 nights in London.

I suggest you cover Scotland and England only. That will be more than enough to take up all the time you have. Here is the route that Rick Steves suggests for Great Britain. (This is also the link that Laurel has furnished.) Take a look at the itinerary Rick gives. It puts you in London at the very end of the trip--Day 21.
https://www.ricksteves.com/europe/britain-itinerary

The only thing I would change about Rick's above itinerary is that it does not give you as much time in London as I would need.
I would take a couple of days away from somewhere else and put them in London.
Needless to say, you do not want or need a car in London.
So if you have a car, turn it in at York and train to London.

You wrote:
"What are the MUST SEE places for each country?"

You really need to buy Rick Steves Great Britain guidebook, and read it from cover to cover.
Then you will be able to create your own "must see" list, which will be tailored to your own interests.
To ask us to compile this list for you for all three places indicates you have done no (or very little) research yet.
I realize this website is the first place people look to for planning, but I would also suggest you read Rick's guidebook, so you will have some "must sees" of your own.
You could create a tentative itinerary and then come back here, post that itinerary, and ask us to critique it. This may be the best way for you to end up with an itinerary you'll enjoy.

With 3 kids along for the trip, you may want to put plenty of castles into the mix.
England has some great castles. Warwick Castle near Stratford-Upon-Avon is one I'm sure the children would like, and the adults also. In the summer, they have jousting and falconry shows, in addition to a very historic castle.
Depending upon the ages of the kids, you may want to include some Harry Potter stuff.
Or Legoland, near Windsor.

Happy travels to you. And happy planning. Doing the reading and research (learning more about each country, town or city on your itinerary) can be the most fun part of planning your trip!

"Should we rent a car, or is train transport good in most areas?"

As several other posters have said, that depends upon where you want to go.

You can put together a really great itinerary for England using only trains for a portion of your trip, then renting a car for just a few days to explore the countryside, such as in the Cotswolds. Then turn in the car, take the train into London.

You can take the train from London up the east coast, stopping at York along the way, continuing on to Edinburgh. After several days exploring the city of Edinburgh, rent a car to explore the countryside of Scotland.

A combination of both train and car can make for a good trip.

England has a very good train system that has covered every place that I've wanted to go for the past 20 trips.
I prefer not to drive because I just love taking the trains. The trains have very comfortable seats (more comfortable than airline seats) and food service on board. You can use the time you are on a train to look out the window at the countryside, read a guidebook, or eat a sandwich. It's very relaxing, especially compared with the stress of driving a car (getting lost, worrying about damaging the car, etc.).
I think your kids will like the train journeys. It's a fun way to travel in England. Especially when you spot a castle out the window of the train!

If I were you, I would drop the Ireland portion, as there is plenty to see in England, Scotland and not forgetting Wales. Ireland is the geographically remote location.

Car or train - how old are your children? If they aren't self sufficient at rail stations (carrying their own bags etc), then a car would be easier. You don't want a car in London. If you are heading from London to York then Scotland, take the train and hire a car once in Scotland. You need a car for rural areas to get the most out of them.

Start in London and end in Edinburgh or Glasgow or vice versa. When are you travelling? August in Edinburgh is festival time, so avoid this, as much of the accommodation will be booked up by now and what is left will be expensive.

Everybody has their own "must sees" and this depends on your interests. Mine certainly would NOT include Stonehenge, the Cotswolds or Stratford upon Avon, although as RS includes them, most on this forum head there.

My must sees include
the Northumberland coast - fantastic, long, wild beaches and great castles plus Holy Island,
Beamish Open Air Museum near Durham,

Eskdale in the Lake District (certainly not the overly busy Keswick),
Duxford Imperial War Museum near Cambridge for old aircraft,
many of the National Trust houses and gardens (short term overseas membership is worthwhile),
York, Whitby and Harrogate plus the North York Moors and the Yorkshire Dales
Rutland - a mini Cotswolds, with chocolate box villages but without the crowds

Scotland
Skye - needs 3 days to do it justice
Ardnamurchan peninsula near Glencoe - a great drive and try to spot otters
Orkney - can be covered as a day trip from John o'Groats
The border towns such as Hawick, Jedburgh etc

My interests may not accord with yours!