Did you also go on to volume 2 of the RTK set? I haven’t done more than glance through volume 1 and read the amazon.com description for the second, but it looks like volume 2 is where he teaches the Japanese readings of the kanji. However, even with mnemonics, it’s still just a system for memorization.
As much as I wish I could just absorb Japanese and learn it effortlessly the way I did with my native language, that’s just not realistic. I think it’s generally recognized that young children’s brains are uniquely able to assimilate language. This doesn’t mean that if your parents didn’t use two languages with you from birth, you’re doomed. It just means that the rest of us have to work a lot harder to get to the same place.
As far as I know, rote memorization is unavoidable. RTK and Wanikani are different ways of making it a little less painful. Some people like to build vocab lists around a certain topic to study related words at the same time. Personally, I enjoy working through a textbook and workbook (currently using Genki) so that I have opportunities to use the words in reading, writing, listening, and speaking. If I can put the word to use, I’m more likely to remember it more easily.
This community has put together a great list of learning resources here:
頑張って！
（がんばって！）