For meaning, it's definitely best to use mnemonics, like those suggested by RTK, and it's also definitely best to learn as much about the original meanings of kanji radicals as possible so that you can whenever possible construct stories (mnemonics) that have something to do with the real etymology of the kanji in question. But you do not have to use RTK/Heisig in order to use that method; there are any number of resources out there that talk about the composition of kanji, teach the meanings of radicals, and suggest mnemonics, for example, The Kodansha Kanji Learner's Course or Henshall's A Guide to Rembembering Japanese Characters.
The RTK method, at least what I've encountered online, is a bit of a pet peeve of mine for two reasons, one of which is what Rusty mentioned, it doesn't teach readings at all (i.e. the website doesn't; the book apparently does list them), and the other of which is order. For example, RTK presents very early on this kanji: 凹. It means convex. Sure, that's easy to remember because it's a picture of the idea it represents, but this kanji is not useful at all to someone just starting to study Japanese. I've been studying off and on for about nine years, I guess, and I believe I have actually only seen this character used in real Japanese one time.
So there are some advantages to the order prescribed by the government, which really does present kanji in an order roughly based on frequency, so when you learn the first 80 kanji for first grade, those are going to be the kanji that are used most frequently (and which make up a disproportionately large amount of all kanji that are used in any given piece of writing). "Convex" and "concave" can wait until later.
Another reason is for tests: if you want to take JLPT or 漢字検定 the kanji will be grouped by grade level.
A third reason is that when you learn kanji in the Japanese (government-prescribed) order, they are grouped by reading. For example:
巧、江、攻、項： These kanji all have an on reading of コウ. Also that they all have 工 as one of the radicals. The kanji 工 can also be read コウ. Whether it was included in those four kanji above for phonetic purposes or not I can't recall, but if you know that 工 is read コウ than you can use that to help you remember those other kanji. And learning kanji in an order that is grouped by pronunciation allows you to see those similarities and notice, "These kanji that all have the same radical in them are all pronounced the same way." That won't apply to all kanji, and it won't teach you every reading for kanji that have multiple readings, but it is still helpful.
That is the main thing I have figured out that is helpful in terms of memorizing kanji readings.
I've also found that it helps me if I not only memorize the kanji itself but at the same time also memorize, in my vocabulary study, some words using that kanji (either the native reading or a compound, for example, for 惨 I already know the word 惨い（むごい） but might add 惨め（みじめ）, 惨殺（ざんさつ） and 悲惨（ひさん） for the readings みじめ、サン、 and ザン). It can be hard to remember kanji in isolation if you don't see them in actual words somewhere.
It is also really helpful for reinforcement if the vocabulary words you are studying to go along with the kanji readings are words that you actually see in authentic Japanese materials that you are reading. But this is hard to control unless you are doing things in the opposite order--you see a word that you want to learn in some authentic Japanese that you are reading, you make a vocabulary card for it, and you memorize a kanji that appears in that word because you don't know that kanji yet. But if you do that, then it is going to be going out of order, if you are learning the kanji in any set order. (Confession: my kanji study has been rather haphazard and I have for the most part not been going in any order, but I have actually had quite good results remembering the kanji that I memorized in the aforementioned backwards order of finding an unfamiliar kanji in an authentic Japanese text I was reading, writing down the vocabulary word to study, and then also specifically studying the unfamiliar kanji for the sake of being able to remember it in the vocabulary word. The problem with this method is, obviously, order.)
P.S. Apparently the way Japanese people learn kanji is by writing them 20 times each, which I think would be very valuable if you have the time to do it. I've been meaning for a while to ask one of my language exchange partners if he has any tips for memorizing readings of kanji (readings specifically, not meanings), since that's usually the most difficult part for me and I don't have a lot of good strategies for it, so I'll come back and post again if I remember to ask him that and he has any helpful advice.
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