I think the trick is to focus on learning vocab rather than just kanji. The issue with just focusing on kanji, is that you just get the kun’yomi and on’yomi thrown at you with nothing at all to anchor it to. It’s like, hey, the on’yomi for this character is しょう, but that’s also the on’yomi for dozens of other characters.

When you focus on vocab instead, you learn the readings in the context of actual words which have actual meaning. As a result, you tend to memorise the readings for the kanji as you go as a natural result. Once you’ve got a bit of vocab under your belt, you can start to intuit the readings for words you’ve never seen before – and sometimes even for kanji you’ve never seen before.

Let me give you an example. Let’s go… 小学校. If you learnt this as individual kanji, the last two are fairly easy – it’s a multi-character word, you can infer that it’s most likely going to be on’yomi, and the last two have only one on’yomi each, がく and こう, respectively. (Well, 校 can also be read as きょう, but you’ll basically never see that.) When you stick them together, there’s a slight tweak in pronunciation – it becomes がっこう (because がくこう is hard to say – you’ll often see this happening with readings that end with a う-sound). 小 also has only one on’yomi, しょう, but it tends to use its kun’yomi in all sorts of weird ways – fortunately this is not one of those ways. So we read this word as しょうがっこう.

That took me an entire paragraph to describe. So now let’s try a vocab-first approach: 小学校 is pronounced しょうがっこう – it means “elementary school”. One sentence and it’s done. At the same time, however, you’ve also been exposed to the knowledge that　小 = しょう, 学 = がっ and 校 = こう. Enounter more words like that, and it’ll sink in more.

So later you learn, say, that 彼女 is read as かのじょ. Some more readings. And later in your trawling around the internet, you come across a word you’ve never seen before: 少女 (ok, maybe you HAVE seen it before, but I’m describing an example, so work with me here =P ). You know from 小学校 that 少 = しょう, and from 彼女 that 女 = じょ, so a fair guess would be to stick those together and go with しょうじょ – and you’d be absolutely correct.

I mean, it’s not a foolproof method – when you come across one of the many exceptions, this method may fail you. That said, when you’ve learnt a lot of vocab, you’ll be able to spot the exceptions as well. For example if you learnt 子犬 is read as こいぬ (and means “puppy”), then come across 子猫, you’re more likely to guess こねこ (which is correct) than, say, すびょう (which is not).

tl;dr version: learn vocab, don’t stress too much about kanji in a vacuum.