Russian has several verbs to express 'studying' and 'learning'.
The verbs учить-выучить are usually used in the sense of learning a skill, or in studying elementary courses. These verbs cannot be used without a direct object.
The perfective выучить means 'to memorize'.
Студенты учат русские слова. The students are learning Russian words.
Она выучила все слова. She learned/memorized all of the words.
The verbs изучать-изучить are used for higher level courses, or when the studying involves detailed examination of the subject. For example, if you are studying a language so that you can speak it, you would use учить-выучить. If you are studying the structure of the language (something a linguist might do) you would use изучать-изучить. Both sets of verbs must be used with a direct object.
The verbs учиться-научиться can mean 'to learn how to do something', and they are used with a verb in the infinitive form:
Я учусь играть в шахматы. I am learning how to play chess.
The imperfective учиться can also be used to mean 'to be a student, to go to school, to attend classes'.
Я учусь в Московском университете. I study at Moscow University.
The verb заниматься means 'to be studying' (what you generally do in the library or at home in the evening), or 'to be doing homework':
Вечером я занимаюсь в библиотеке. In the evening I study at the library.
The verb заниматься is also used to indicate someone's profession. The profession will be in the instrumental case:
Она занимается математикой.
The verb pair учить-научить is used to indicate that someone is teaching someone else to do something. The person taught will be in the accusative. The subject taught will be in the dative case.
Наташа учит студентов русскому языку. Natasha is teaching the students Russian.