All citizens of the EU and EEA countries have the right to work in France without a work permit.
A Finnish citizen living in France have the same rights as the French citizens with regard to working, work conditions, accommodation, vocational training, social security, and trade union membership. If you are working alongside your studies, you often are transferred to the social security system of the country in which you are working. Remember to always inform Kela about any work you have done abroad.
To begin your job search in France, you may want to take a look at the EURES Platform. It contains information about living and working conditions in different EEA countries, as well as about the job markets in different parts of the each countries.
In France, just as in other countries, personal networks are valuable. While job searching, it is useful to be active and, if possible, to personally visit prospective employers.
While you are still studying, you have the right to work up to 17 hours per week. Yet, do bear in mind that the studies in and of themselves are demanding and require a lot of time.
To look for jobs in your region, start with job announcements at your university.