учиться, учить, выучить, заучить or whatever.
The root of them is "учить" (to study or to learn in English), and these verbs differ only with the prefixes (вы-,за-,на-). Each different prefix slightly change the meaning of its verb.
"выучить" and "заучить" show some final results you have succeeded while studying.
Я выучил/заучил пять новых слов.
I have learnt/memorized five new words.
"изучить" means you have obtained essential knowledge by researching some subject of your interest, not indicating the steps you have taken to achieve that. "Изучить" sounds more seriously than simply "учить" I would say, scientifically.
*Я изучил устройство кофеварки от А до Я.
I have learnt the operation principles of a coffeemaker from A to Z.
За два года я изучил грамматику русского языка.
It took me two years to master Russian grammar.* (No need to say that you had to learn words and grammatical rules, you've just done it as a whole and that's all.)
"Научить" means teaching something to someone and definitely having result in that.
*Он научил меня рисованию комиксов.
He has taught me how to draw comics.
Он учил меня рисованию комиксов.
He was teaching me how to draw comics.*
"Заниматься" is idiomatic, it says that at the moment you're busy with something. If you omit the reason of your occupation, then studying is implied.
У меня нет времени, я занимаюсь.
I've no time, I'm studying.
"Обучаться" or "Обучиться" indicate professional training.
Их обучали кройке и шитью.
They were being trained to be a dressmaker.