How can I travel from Chamonix to Milan?

Alex,

Check this website to search all you transportation options: http://www.rome2rio.com

It looks like your best option is to take a bus.
You can also travel by train but this option will require multiple transfers.

Hope this helps!

The obvious answer is train. According to www.bahn.de there is a connection:

Depart Chamonix-Mont-Blanc 12:54, arrive Vallorcine 13:30
Depart Vallorcine 13:43, arrive Martigny 14:35
Depart Martigny 14:40, arrive Sion 14:53
Depart Sion 15:15, arrive Milano Centrale 17:37

These are times for May, times for July are not yet posted, but they should not change much.
Your question is ambiguous as to whether you want to know how to get to Chamonix as well. But, since you don't say where you are coming from I can't answer that question.

Sometimes, you can go over the mountain via the Aiguille du Midi gondola, but the Italian portion of that route (from Hellbronner to La Palud) is closed in 2015.

In summer, buses depart from Chamonix train station to run 4x/day through the Mont Blanc Tunnel to Aosta, Italy (2-3 hours), see www.sat-montblanc.com; you can catch an Italian train at Aosta and the town also has some Roman ruins. The tunnel portion is 7 miles with no scenery.

These options are both discussed in the Chamonix chapter of Rick Steves' France guidebook, but not in the Italy book.

If it were me , I would go the route suggested by Chris for the following reasons -- I dislike bus travel and given the choice of rail , it wins for me every time . I'm a rail fan , so I am also prejudiced in that direction . As has been said , the bus might be a shorter trip , BUT , the route by rail is the Mont Blanc Express which carries you up into the mountains past picturesque villages perched high above the Trient Gorge on its way to Martigny , Switzerland for the second of the three transfers on this trip . The vistas are stunning ( sit on the right side of the train , overlooking the valley and drool ! ) . I pack VERY light , so making changes is a non - issue for me ) This trip was one of the highlights of my last visit to Chamonix . http://youtu.be/xhLj35U-igY and the TMRSA site http://www.tmrsa.ch/cms/en/mont-blanc-express

Alex,

I may also be in Chamonix later this year, so have been researching the same question. Although my route will be slightly different, I plan use the same solution that Chris posted.