Go to a Museum. The Alpine Museum in Chamonix is well worth a visit. You can also take a tour down in Sallanches and visit the Mountain Nature Center.There is also the Galerie du Globe in Argentiere as well as art exhibits in many of the hotels. Further afield, the Art Gallery in Martigny is excellent.
Head to the Cinema Vox which plays the latest films from around the world. A selection of French, English and other foreign films (all subtitled in French) on three screens. Look for the "vo" or "version originale" (original audio) next to the title to find which films are in what language.
The Chamonix Library is a great place to spend an afternoon. For opening hours: 33 (0)4 50 53 34 82. The Library often has storytelling and various activities for children.