Is it necessary to purchase a ticket here in the States or can we just buy one when we get to the Milan Centrale station?
You need to provide more information. For example: When will you be travelling? Night or day train? High speed or regional train? How far in advance will you be in Milan? First or second class?
If you take a direct EC Cisalpino train that goes all the way to Geneve, you certainly can buy a ticket in Milano. If you have to change to a Swiss train in Brig, you may need to buy a ticket for Brig-Geneve there.
A general answer to your general question is, YES, you can and should have no problem buying a ticket after you arrive. Answer Jeff's question if you want a more precise answer.
Jeff - we'll be traveling in the day - Thursday, April 2nd around noon. I believe there is a 12:30 train Milan,centrale to Geneva. 2nd class. Not sure if it's a high speed train - don't think so. It's only about 17E for that time of day.
It is not necessary to buy the ticket here. Buy it there.
Theresa,
The standard 2nd class walk-up fare for that train is €59. The fare you quoted is an advance purchase discount fare. Since the Trenitalia site doesn't accept North American credit cards, book your tickets on the Swiss Rail site and have them mailed to your home. That site doesn't show the discount for 2 April, but it does for dates in March. Keep checking the site until the discount fare appears as one of your choices.
The actual discount fare is CHF 32.00 (€19.51).