A thorough answer to your questions would require a very long post. I will only give a partial answer that I hope helps somewhat.
I recommend learning the characters in roughly the same order that Japanese children learn them. That order basically results in more commonly used characters being learned earlier than less commonly used characters, although there are a few notable exceptions.
One resource I used and liked was the Kanji Gakushu Step (漢字学習ステップ) books, designed for Japanese children. This is a series of 12 different books, whose levels correspond to the levels of a set of tests of kanji ability called the kanji kentei. The introductory book (i.e., lowest level) is level 10, which covers the 80 characters taught to first-year elementary students - ostensibly the 80 simplest and most commonly used characters. One drawback to using these books is that everything is in Japanese, so, for example, you need to learn some Japanese just to understand the example sentences and phrases, which may seem like a catch-22. Using these books results in a high front-end loading because of this, and so this approach is not for everyone. However, the level 10 book is written for Japanese children, so most of the example sentences are suitable for someone in the early stages of Japanese learning, so you would not be wasting your time studying content that is too far beyond your ability level. And studying this way might help you open your mind to gaining an understanding of how kanji work. Part of learning kanji is not just learning individual characters themselves, but gaining a feel for how kanji work, as there is simply nothing like them in a language such as English.
I don't recommend the Heisig book(s), as, among other complaints, I think his mnemonics are too silly and not easily remembered without much effort. Also, he presents characters in an order very inconsistent with my earlier recommendation. However, some people swear by his books (actually, notably, the first of the three books is far more popular and respected than the others), and there is an online community and a number of support resources tied to his method. As for the claim that his key meanings are inaccurate, I don't know whether or not that is true, but I know that he boils down each character to a single English meaning, which I do not think is a good idea. I think that it is better to gain a sense of characters than to rigidly tie them to a single English word or phrase.
As for the memorization that is required for learning kanji, many learners recommend using a spaced repetition system (SRS). Two popular apps which utilize SRS and whose users have made kanji databases available are Anki and Memrise. Using one of these would not be a bad approach.
There is another popular online SRS program which I do not care to name. I don't recommend it, because it is tied to a monthly subscription, and designed to keep you subscribing as long as possible. Plus, it oversimplifies some of the content it teaches, and uses non-standard terminology in keeping with the image it tries to convey that leaning kanji is easy. It has many users, though, some of whom swear by it. For me, though, kanji is not something that was intended to be learned with a subscription.
As for your question about learning one or all meanings, it depends on the character. Some characters have two or more very commonly used meanings. For example, depending on the context in which it is being used, 月 can stand for the moon, or for "month", or even for Monday. Even a beginning learner should be aware of all of these meanings. Other characters have one quite common meaning and other more obscure meanings. For such characters, only the most common meaning needs to be learned at first.
I want to learn kanji. I was using Heisig's book but I've since heard that the key meanings are inaccurate? Can anyone comment on this? So I'm looking for advice on where best to learn the kanji themselves and how to use them/the grammar etc. Any recommendations of sites, books, other resources are welcome. And can anyone tell me if I really need to learn all the possible meanings of a character or just one? Please help!
