General:
* Learning French
* Trying to learn French
* Studying French
* Taking French lessons
* Improving my French
Emphasis on effort:
* Struggling to learn French
* Working hard to learn French
* Making an effort to learn French
* Putting in the time to learn French
Emphasis on progress:
* Beginning to learn French
* Getting started with French
* Making progress in French
* Learning French bit by bit
More formal:
* Engaging in the study of French
* Pursuing French language acquisition
The best way to say it depends on the context and the specific message you want to convey.