The verb изучать is used to indicate the subject that is taking at university, college, school, etc.The subject studied is the direct object of the verb изучать. It must be in the accusative case and must always be mentioned. A historian is studying an ancient manuscript. - Историк изучает древнюю рукопись. She studies medicine. - Она изучает медицину. She studies the origin of light. - Она изучает происхождение света.The verb учиться is used to indicate that someone is a student, i.e. goes to school, university, college, etc. It is also used to describe how well one is doing at school, university or college. This verb never takes a direct object. Не studies at the medical college. - Он учится в медицинском колледже.I study well. – Я учусь хорошо. The verb учиться is also used to denote human activities and actions: She learns to swim. - Она учится плавать. They learn how to drive a car. - Они учатся водить машину. The verb учиться (to go to school, to learn) must not be confused with the verb учить (to learn, memorize). These are two different verbs. She is learning English now. - Сейчас она учит английский. Have you memorized the multiplication table? -Ты выучил таблицу умножения? The second meaning of the verb учить is to transfer knowledge and skills, to teach, instruct, give lessons.She was educating her children by herself. - Она сама учила своих детей. I'll be teaching you the Russian language. - Я буду учить вас русскому языку.The verb заниматься is used to describe the process of studying (doing homework). This verb never takes a direct object. I often study (do my homework) in the library. - Я часто занимаюсь в библиотеке.

Oops! Sry, wrote the answer in comments. Учиться is like just "studying" - the process of learning. But for example, if you are doing some researches in something, you can say "Я изучаю". Заниматься is just an action of doing something. If you are tidying your room, for example, you can say "Я занимаюсь уборкой" as well. It's like "I am doing the cleaning of my room", something like that :D So "заниматься" you can use with any action, not only studying. However "учиться" can be used only when you are learning something. "Учиться рисовать"(learn to draw), "Учиться ходить"(learn to walk), "Учиться играть на пианино"(learn to play the piano) etc.

Учиться is like just "studying" - the process of learning. But for example, if you are doing some researches in something, you can say "Я изучаю". Заниматься is just an action of doing something. If you are tidying your room, for example, you can say "Я занимаюсь уборкой" as well. It's like "I am doing the cleaning of my room", something like that :D So "заниматься" you can use with any action, not only studying. However "учиться" can be used only when you are learning something. "Учиться рисовать"(learn to draw), "Учиться ходить"(learn to walk), "Учиться играть на пианино"(learn to play the piano) etc.