As far as Japanese is concerned, I agree with the principle that you should learn vocabulary in context rather than learning kanji characters for their own sake. If you follow a textbook learning approach then you will be introduced to more and more kanji as you learn more vocabulary, so it's not a "one or the other" situation.
Even so (and hopefully I'm not contradicting myself too much), I decided fairly early on to try to learn the core meanings of all the Joyo kanji. I think this is not for everyone (it took me about 18 months), but for me it seemed worth the effort. It obviously enables you build up a good familiarity with all the kanji, so that everything doesn't just seem like an impenetrable forest of unknown symbols. Also, if you set up a file of characters along with their core meanings (in English) and readings, you can very quickly locate any character you encounter by searching on the meaning that you know, and then easily copy and paste the character into a dictionary, and/or check the readings, or whatever you want to do. I personally find this so much quicker than looking up a character by radical/stroke-count, or trying to draw it in some kanji recognition program, that I think this one technique has actually saved me more time than it took to memorise the meanings in the first place.
Even though I have memorised the core meanings of many characters, I find that when I'm reading fairly fluenty, which I can currently manage only with very simple Japanese, I do not consciously refer to that memorised knowledge. Instead, I see the kanji as words, or parts of words, i.e. as vocabulary. When I hit something kanji-related that I can't read fluently (which is much of the time!), I can try to glean a meaning from the recalled core meaning, and I can quickly look up more about the character in the way I described. Eventually when I can read even advanced Japanese fluently (if I don't die or give up first), I will no longer need that rote-learned knowledge.
My original plan was to rote-learn character readings too, but after several attempts I gave up. I found the readings completely unmemorable in isolation, and the whole exercise was just mind-numbing. Now I treat all readings as vocabulary learning. So, I aim to learn the pronunciations of words (sometimes single kanji, but often kanji compounds or kanji + okurigana), not character readings in isolation.