To begin with, DON'T use Rail Europe as they're not a reliable method for searching rail routes. Use the SBB site mentioned in the previous reply.
For the trip from Milan to Chamonix, according to the Bahn.de website your best bet is probably a departure from Milano Centrale at 08:23, arriving Chamonix-Mont Blanc at 13:10 (time 4H:47M, 3 changes, reservation compulsory on first segment which will be provided with the ticket). That will provide a bit of a "cushion" in case of train delays or whatever. There's an earlier departure at 07:23 if you'd like to arrive at 12:10.
If you buy tickets in Italy, they may only be able to sell tickets for the first segment from Milan to Sion, and you'd have to buy tickets locally when you arrive in the other stops. The SBB (Swiss rail) website should be able to sell you tickets for the entire route, but I haven't checked. Although your destination is in France, you'll be travelling through Switzerland to get there.
As the first segment from Milan requires reservations, you might want to purchase tickets in advance on that train. You should be able to purchase tickets for that at Trenitalia, CapitaineTrain.com or ItaliaRail.com. The tickets will be specific to that train, date and departure time, so choose carefully as they will likely be non-refundable and non-changeable.