* J'apprends le français. (Formal and most common)
* Je suis en train d'apprendre le français. (More emphatic, meaning "I am in the process of learning French")
* J'étudie le français. (More formal, often used in academic settings)
* Je prends des cours de français. (Literally "I take French lessons")
The best way to say it depends on the context and the level of formality you need.