We have been lucky to live in large, multicultural cities. My kids attended "Saturday school" for years to learn Japanese. We also belonged to the cultural centre which sponsored courses, martial arts classes, festivals, movie nights etc.
Cooking is fun - you can research a recipe that is a local "specialty", learn about the agriculture of the region - and the geography (rice needs wetlands etc.), take a trip to Chinatown for supplies (if you have one nearby), and have a great meal at the end of the experience.
Resources - internet, of course. Libraries. A subscription to a foreign children's magazine - often easier than books because the articles are short and they have pictures. Movies and DVDs - use the English subtitles to help!
Your children have a great advantage, if your DH has taught them the language. My father didn't teach us Japanese because the emphasis was on assimilation in North America after the war. I regret it a great deal, although I understand why he didn't.
I think it helps if both cultures are part of daily life - language, food, habits (shoes off indoors!). If you don't speak the language, you could learn some greetings and useful phrases and use them daily.
We are planning a trip to Japan next year - that will be our ultimate learning experience!