You should use something that is aimed at second language learners, not Japanese children. An elementary school student doesn't know much kanji or formal grammar, but they are fluent in their own language. Therefore, the things they are taught in school are different from what you need to learn.
Erin's Challenge is an online course provided by the Japan Foundation (who run the JLPT), you might want to check it out. I think it's probably the best of the free online resources.
Genki is one of the most popular textbooks for self-studiers. It can be a little pricey - I suggest looking for used copies of textbooks or checking your local library in case there is something available there. At the beginner stage, don't bother buying lots of things like reference books, manga that you think you would like to read in Japanese, etc - there is plenty available online. Longer term, you might want something like a grammar reference - I've heard good things about the Dictionary of (Basic/Intermediate/Advanced) Japanese Grammar series but don't own them myself.
You should try to set aside some time every day to study; regular, consistent study is the most important part of learning a language. There is not a single best way to learn - it depends on how you learn best, what materials you have access to, what your priorities are (reading/writing/listening/speaking), etc. Everyone's experience is different - things I think helped me have been:
1) Using my commute/walk to work (downloaded a bunch of learning podcasts; half an hour in the morning, half an hour in the evening, every day - it adds up).
2) Focusing on my areas of interest (learning specialist vocabulary for the things I wanted to read about)
2a) Not caring about lists of kanji and vocabulary which are
not related to my areas of interest (jouyou kanji, looking at you)
3) Not being afraid to attempt to read - and to abandon reading material if it turned out to be too far above my current level.
4) Buying a second-hand DS and a bunch of Japanese kanji games on ebay.
5) Switching much of my online time wasting to Japanese websites so I am at least wasting time in a different language.