Characters are a bitch, no way around it. Your kids will have to dedicate a large chunk of their time to learning reading and writing in Chinese. After that it's a continuous chore to retain that knowledge, especially in writing. After several years study, it can seem like you're set to the Sisyphean task of building a mountain out of sand--focus on building up the peak with new knowledge and other memories decay. That said, there are a billion plus living examples it can be done, and there are things that can certainly help. Just don't think it will be easy.
With Chinese it's kind of hard to dive into new reading material. You either know a character already, or have no clue what it means or even how to pronounce it. That, and every character being unique, means reading/writing will be the limiting factor in your kids' language study and the most time-consuming to remedy. Below are some tips to break down the task.
First thing is to learn the radicals. There's a limited number of them, and at least one in every character. Learn how to draw them because they're used over and over again. Learn their meanings too, because a character's meaning is usually at least loosely tied to its radical. Learning to identify the radicals also helps greatly in looking up unfamiliar words, as Chinese dictionaries are traditionally arranged first by radical, then by number of strokes.
When you buy them a dictionary, get a beginner's dictionary so that they can have a larger font, usage examples and Pinyin pronunciation, all of which are sometimes missing in comprehensive dictionaries. A good choice that provides many example sentences and phrases would be The Starter Oxford Chinese Dictionary (sorry, Simplified version only). Get them a second dictionary later on if they can't find every word they need. For several reasons, I like Chinese Characters: A Genealogy and Dictionary. You can try out the online version of it at Zhongwen.com to see how it's organized. This is also the only dictionary that you can use by looking up any part of a character, not just the radical (which can sometimes be hard to identify).
Many characters are comprised of radical-phonetic pairings, where the non-radical part hints at the sound of the word. They'll notice many more of these related character components at the intermediate level. However, given the ~4,000-year development of the written language, these links can often be tenuous. Thinking up elaborate stories trying to tie all the pieces of the character together can be quite useful. For instance, with the character for wrong () I remember it by thinking, "It would be wrong to bet money that sun sets underground." A little convoluted, but it was enough to jog my memory ever since. Useful as this strategy can be, it's just not always possible and you'll have to learn many words by rote memorization.
For this I recommend writing. A lot! Have your kids say the words aloud and think of the meaning as they write. After enough repetitions, hopefully it will become part of their "motor memory" and once started they will be able to finish a character almost by reflex. They'll need this level of ingrained familiarity if they hope to retain the knowledge for long.
It's essential then to review regularly and for them to brush up on what they forgot. Flashcards can be used as others suggested, but I'd recommend using a "3-sided" flashcard that shows the English translation, the character and the pronunciation all separately. You can do this by writing along the top and bottom of one side of the card and holding them so you don't see both at once. This way they won't depend on the Romanized pinyin to pronounce characters. To optimize learning, reorganize the cards based on how well they're known. This way time won't be wasted needlessly reviewing stuff that's already learned.
To help with this optimization, some people use computer programs to model their memory decay, bringing up the character flashcard only when it's likely to be on the verge of being forgotten. SuperMemo is the most famous commercial software, and comes in versions that can run on PDAs, so you'd have a stylus for practicing writing on the go. On PC/Linux/Mac the free, open-source Mnemosyne attempts to replicate SuperMemo's method and has extensive libraries of Chinese cards available for download. I haven't used Mnemosyne extensively, but it looks very promising for memorizing lots of data--of any sort. You can do true 3-sided cards and even place audio files in them.
To get outside of the dreary task of studying, try to locate some childrens reading material from China (or Taiwan if they're learning traditional characters). Books for very young kids will include the pinyin or bopomofo pronunciation alongside. By young grade school, the books only include that for the new or hard words, and soon no hints at all. If you can find comic books, that will probably be a whole lot more interesting for them. I'd also recommend finding an online language partner if they have no other native speakers to practice with. Sites like The Mixxer and E-Tandem can help them find partners to chat with through Skype. This will help build confidence in actually using vocabulary, which is the only way stuff will really stick.
"Little prigs and three-quarter madmen may have the conceit that the laws of nature are constantly broken for their sakes." -- Friedrich Nietzsche