I have a multiple-entry Italian Schengen visa that will last a couple of months. Within this period, I want to go to Switzerland for a weekend, and to Austria for another weekend. Can I just get on a train and arrive there? Or do I need to do some paperwork, request another visa, or something?
Yes, you can just go on a train and arrive there.
A Schengen visa allows you to travel anywhere in the Schengen region. It should be even less trouble with a multiple-entry visa, where it is expected that you will use it for trips in addition to the one you originally applied for.
The Schengen construction is currently creaking a bit under the weight of the refugee/migrant situation, and there's been some talk of "suspending Shchengen" that may sound alarming. But this is just a matter of introducing systematic border checks at some of the internal borders, and does not involve changing the rules for who is allowed to go where. There's no talk of changing how Schengen visas work anywhere in the near future, much less the validity of already issued ones. You might experience having to show your passport/visa when you cross the Italy/Austria border -- but since your visa is actually in order, that should not be a problem for you.