One of the only ways to get an internship (known as a stage in French) is to have a convention de stage. This is a special authorization from a university that gives you the right to work for credit as a student. Many companies will not even look at your C.V. unless you are in a university program. Therefore, you will need to enroll in a language school at the very least to even be considered for an internship in France. Luckily, school in France never costs more than five hundred euro a year. If you need a visa and a convention de stage, it will not cost you very much compared to universities in other parts of the world.
The average salary of an intern in France is roughly four hundred euros a month. You will be working all day, every day just like an average worker at your work placement. This might not be a large sum of money, but it is better than working for free like most North American internships! Plus, you will still have all your basic access to the housing allowance (CAF) and state medical care.
If you do not have any kind of access to a university because you have graduated, then reach out to alumni or your alma mater itself. They might have connections to companies that will hire you for an internship in Paris or a larger city. This will let you work, earn money, and learn French while pursuing your dreams.