It doesn't matter if you're travelling by train, plane, or automobile, vacations with children are anything but cheap. Going in the off seasons may save money, but it also has its downsides. Besides the problem of removing children from school, weather might be less than ideal and some activities geared toward children may be closed. Finding the "best" time to travel with children is more complicated than just nailing down a time of year.
After a long school year, Summer break is clearly the most desirable time to travel. But while it may be the best for scheduling, it's not necessarily the best time for your pocketbook. During the middle of Summer, some hotels and flights can be more than double their standard fare.
If summertime travels are your real goal, the closest you can get to the end and beginning of school (early June and late August) may be the best. At this time, hotels haven't reached their peak pricing and the weather is most likely going to be tolerable. While this may mean having to pack your suitcase the same day you're also unloading their backpacks, it could save you a ton of money.
Outside of the Summer months, the cheapest times to travel are January and February. Many kids have a break around Christmas, so it may line up that the time just after New Year's is the best time to fly. Yet that doesn't give families a lot of time, and thanks to a little thing called "snow," it limits where people want to travel.
At the end of the day, travelling with kids is going to require some flexibility. Finding long weekends in the off months can help save money but might have other complications. Picking the best time to travel for your family requires a balance of schedules, locations, cost, and weather. Of course, watching their faces explode with joy at seeing the beach for the first time or their favourite Disney characters makes it all worth it.