Susan, you might find it helpful to read Rick's "Rail Skills" chapter in Europe Through The Back Door (I would HIGHLY recommend it!). It helped me a lot to understand the rail system prior to my first trip to Europe.
Regarding tickets and reservations, most of the "fast" trains (TGV, etc.) require compulsory seat reservations. If you buy P-P tickets, the reservation fee will be included (as opposed to using a Rail Pass, where this fee has to be paid separately). Unless it's a major holiday or something in Italy (or there's a strike happening), you should be fine buying tickets in Milan for your journey to Vienna and then to Trieste. I usually try to buy tickets a day or two before my outgoing journey, and so far haven't had a problem with that approach.
Since this is your first trip via train, you'll need to give some thought to which train you choose and the number of changes. You didn't indicate your travel dates or the time your flight arrives in Milano, so it's difficult to suggest a particular train. One example though is a train departing Milano Centrale at 07:05, arriving Wien Westbahnhof at 18:28 (Time 11H23M, one change at Innsbruck Hbf). Note that many cities have more than one station, so that's an important consideration also.
I've never flown into Milano, so not sure on the method to get from the airport to Milano Centrale. If you could provide more details, I'm sure the group here would be able to provide some good information.
Cheers!
