That said, I've found that the kanji separately really helped my Japanese studying. If nothing else, it's kept me from falling too far behind the Chinese and Taiwanese students in class...
The book I used for the kanji is The Kodansha Kanji Learner's Course by Andrew Scott Conning. Can't recommend it enough.
(Though, it should be said that the reason the Chinese kids are better isn't that they "know kanji", but that many of the words are written the same way in Japanese. Substitute some of the kanji with hiragana, and they'll lose the bubble quickly.)
The only things I can say with confidence are:
* focus on words
* learning words becomes faster in *any* language as you gain proficiency
* learning radicals helps you recognize kanji, and is a good thing to do