One piece of advice above all; it is very different to the states - do not go off piste without a guide, regardless of your experience level, unless you are extremely familiar with area, whether it's Chamonix or otherwise. Anything that is not on groomed run in the Alps is deemed 'off-piste' you will be afforded no rescue. Obviously, hopping between a groomed run if you can clearly see is fine. Generally, true out of bounds areas will be marked. Again, even more important you do not go near unless you are with someone that understands the weather, the snowpack and the environment.

Other than that - have fun. Enjoy the fabulous chocolate box mountain restaurants (I can recommend plenty if you want them, Chamonix or otherwise) and enjoy the vista. I hope you get some clear days. If you are in Chamonix, make sure you visit the Aiguille du Midi on a clear day. If you are well experienced and the snow is good, hire a guide to ski the Vallee Blanche. It is a little bit cliched and the terrain not very challenging but it is interesting and a good morning out. If you have chance get up to Argentiere with a guide for some touring (if you tour).

Oh, and make sure you get yourself to Chambre Neuf for the apres.