Almost every student asks me this question. And this is what I tell him/her:
ИЗУЧАТЬ means “to study” (in depth), to research. It is used when we say what we study at university: Я изучаю математику в университете. It means that you study maths hard, you read many books about maths, you write essays and thesises.
УЧИТЬ is “to learn” (memorise). You can learn грамматику, новые слова, стихи and so on. So you read it, look at it, practise it many times in order to memorise it.
УЧИТЬСЯ means “to learn” or “to study”. The difference with the previous two is that you can not use a direct object in Accusative case after verb “учиться”. The biggest mistake is “Я учусь русский язык“! We use this verb to answer the question “Where do you study?”: Я учусь в университете. Or “What are you doing right now?”: Я учусь.
So if you learn or study and you just want to say what you’re doing, about this action itself without saying specifically what exactly you learn, use УЧИТЬСЯ. If you study something in depth, research the subject, use ИЗУЧАТЬ + direct object in Accusative (математику, физику, химию). If you want to memorise a rule or a poem or want to say that you learn Russian (but not that serious as they do at universities), use УЧИТЬ.
Any questions?
