This could take some time.
Travel is very different from just going to a place to look it over, I have done both. No, it does not have to cost a lot.
The motorhome idea is a good one, this can be a self done conversion while you are planning your trip(s). Forget the old VW campers and get as big as you can handle, a web search will give you lots of encouragement. One of the drawbacks of Motorhomes is getting them from continent to continent, usually expensive and a bit of a risk. People do do it though and it still appeals to me.
Backpacking is not a good idea with small children, they will soon get fed up.
Working while you go is a bit of a gamble when you have your childrens well being at the top of your list, you will accept roughing it more than they will. You also have to keep your childrens education in mind. While a trip of any kind is an education for children they still need the formal bit.
Have you done any travelling?, if not start with a trip to Europe or North America. These places have lots to see and have the trappings of our culture. There are cheap flights to America and you can hire a car, or buy an old truck, throw some camping gear in and take off. An example of this type of approach is to be found on
http://www.petaflop.de, look for their USA/Canada tour. Be safe about camping in a country that has wildlife that will kill you, such as Western Canada.
I worked with a guy that once did a several week African safari in the busses that they use, he met people in the travel industry and was offered good deals for years after. This is not practical for you as you have to pay for Four, not one.
Your best bet seems to be the car/camping or motorhome. You can drive from France to South Africa via Russia, China, Turkey etc. That is however, a long arduous trip. You could plan Five week trips, on the cheap of course, for the next few years, subject to your work, and take them in the school holidays.
Travel can always be done cheaply, that is sometimes the most enjoyable way to do it. Always look for the cheapest option and make the most of it. As an example, in the USA and Canada trains are a bit pricey, if you look at their specials pages you can save lots on most of their routes, you may need to modify your plans though.
Should I ever get the time and money again I would be motorhoming across Canada and Alaska for a few weeks.
Plan it loosely but plan it safely.
Addition.
I forgot, look on the 'lonely planet' site and join the 'thorn tree' forum.