I'm also looking for this kind of material: mainly Japanese texts with furigana and an English transcript to check I understand correctly.
It's harder to find than I thought.
So far I used Hiragana Times (somebody probably talked about them already on the forum).
I also found Nippon Talk - Japanese texts with furigana and English translation. their content is free.
Jコミ - not everything here has furigana, it varies a lot, but some have furigana for just about every word. This is a legal/ad-supported site, the manga are generally older things from the 1980s/1990s. There's also some light novels.
ETA: oh ha I didn't even see the date of the original thread. whoops.
Feel like beeing late, but who cares... 昔話のジャンル別 福娘童話集
I find this page really good, even though there's no furigana. You can listen here japanese stories for children and it's quite amazing. I found out about it just today and I hope it will help me