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
Question 1: How much will the ticket cost?
Question 2: What is the name for the specific train?
Question 2: What is the name for the specific train?
Trains don't have names they have numbers. The same result that you copied above also has the train numbers.
Depart Chamonix-Mont-Blanc 12:54, arrive Vallorcine 13:30 is TER18913, a French regional train
Depart Vallorcine 13:43, arrive Martigny 14:35 is R 26213, a Swiss regional train
Depart Martigny 14:40, arrive Sion 14:53 is IR 1821, a Swiss inter-regional train
Depart Sion 15:15, arrive Milano Centrale 17:37 is EC 39, a EuroCity train jointly operated by the Swiss National Railway
(SBB/CFF/FFS - the SBB is the German language, the CFF is French and the FFS is Italian language abbreviations for the same national railway system) and the Italian one (Trenitalia). The train coaches might be Swiss or might be Italian, but are most likely Swiss. That train originates in Geneva, so if you took the bus to Geneva from Chamonix you might be on this very same train from its origination.
Question 1: How much will the ticket cost?
When? How many? Ages?
Once you know that you can use the SBB/CFF/FFS website at http://www.sbb.ch/en/home.html and it will also tell you how busy each train is likely to be and the prices. Bear in mind that the headline price on that website usually assumes that you have a Swiss Half Price card - most Swiss people do - so make sure that the price you get does not include that 50% discount - unless you do have a discount based on a pass or half fare card that you have purchased.
Fares are unavailable at the moment due to construction, in the Chamonix.
I estimate that second-class tickets for the whole route wouldn't cost more than the equivalent of $85 US per adult, and could be less.
An easier solution would be to take one of the Chamonix shuttles to the main station in Geneva. I haven't checked on times, but if you're able to get to the station at about 07:15, there's a direct departure to Milano Centale at 07:39, arriving 11:37 (time 3H:58M, train EC 37). That will be much easier and quicker than the one with multiple changes on the other route.
