Can I go through Italy first with an Austrian visa, or do I have to go through Austria to go to Italy?

There is no problem in principle, but if you take a fictitious travel itinerary and get a Schengen visa from Austria with a hotel order, and then you have never actually been to Austria, or your stay in Austria is shorter than that in Italy, then the visa officer may be suspicious when you apply for a visa next time.

In short, it's not a big problem, but since I've been to Schengen, it's easy to sign Italy, so why bother, right?