Your first address for learing French is most likely the University’s Center for Fench Studies, which offers courses for foreigners at all levels.
If you are a PhD student, French courses can also contribute to the mandatory training hours. Check the list of courses currently offered by the Doctoral School here.
If you prefer to improve your French in a more relaxed and personal setting, « Association IntEGre » proposes tandem sessions (« Café Tandem ») regularly during the semester. Check out their Instagram page for more information.
Finally, Alliance Française de Grenoble proposes a courses and certificate exames.
The CNRS can take care of French courses that will take place at the Institut Néel.