For people who know some Japanese and start trying to read, I recommend the Japanese Graded Readers. They go from level 0 to 4 and contain small booklets with furigana for each kanji. They are designed to be used by non-Japanese without a dictionary and come with a CD. They can be bought individually or as a complete set. https://shop.whiterabbitjapan.com/products/the-complete-set-japanese-graded-readers
Once you have read all the readers, I suggest the "10 pun de yomeru ohanashi", "10 pun de yomeru meisaku" and "10 pun de yomeru monogatari" series. They are three sets of 6 books from 1st grade to 6th grade and are aimed at Japanese children. Each book contains stories, fairy tales, poems and sometimes non-fiction. They have a lot of furigana in the first grades, then less and less. I have three sets and am going slowly through the 4th grade books from each set. https://shop.whiterabbitjapan.com/pages/search-results?q=ohanashi&p=1http://www.amazon.co.jp/dp/4058110120 http://www.amazon.co.jp/dp/4058111917
Similar series I haven't tried are the "Ikki ni yomeru" and "Kyôkasho ni detekuru" series. https://shop.whiterabbitjapan.com/products/ikki-ni-yomeru-complete-sethttp://www.amazon.co.jp/dp/4591091678
Other children's books I like are the beauitiful album "Tebukuro o kai ni" (with furigana), the poem compilation "Kodomo to isshoni yomitai shi 100" (without furigana - good for practicing kanji with very short texts!) and bedtime story compilation "Atama no ii ko o sodateru ohanashi 366" (with furigana - written small, for good eyes only!). If you are fan of mangaka group called Clamp, "Soeru to Larg" is a very cute album.http://www.amazon.co.jp/dp/4039633105http://www.amazon.co.jp/dp/4901364162http://www.amazon.co.jp/dp/4072795682http://www.amazon.co.jp/dp/4063645959
I learn kanjis using "A Guide to Reading and Writing Japanese", a book containing kanjis in the order Japanese Children learn them instead of the order non-Japanese usually learn them. That means I learn kanjis in sync with reading the "10 pun" series. The book makes it easy to photocopy each page after it is learnt and making kanji cards out of it. http://amzn.com/4805311738
For learning grammar and vocabulary, I liked the "Minna no nihongo" series. I went through the beginner 1 and beginner 2 levels, and you can go far with those. They contain theory, exercises and a CD. The textbook and translation/grammatical notes book are necessary, the other books in the series are optional. https://shop.whiterabbitjapan.com/products/minna-no-nihongo-shokyu-1-honsatsu-textbook-for-beginners-second-edition https://shop.whiterabbitjapan.com/products/translation-grammatical-notes-for-minna-no-nihongo-shokyu-1-second-edition
White Rabbit Japan is my favorite seller. I also like Yesasia. Amazon.co.jp has the most choice, but shipping is most costly.