I am a USA citizen. My wife is a Chinese National with USA Permanent Residence status. We wish to travel to Italy for vacation. What do we need to do?
Here's the relevant requirements from Timatic:
Passport required. - Passports and other documents accepted for entry must be valid for a minimum of 3 months beyond the period of intended stay.
Visa required, except for Passengers with a Hong Kong (SAR China) passport for a maximum stay of 90 days. (SEE NOTE 56322) NOTE 56322: The max. stay is granted within 180 days. Visa required, except for Passengers with a Macao (SAR China) passport for a maximum stay of 90 days. (SEE NOTE 56322) NOTE 56322: The max. stay is granted within 180 days.
Unless she holds a Hong Kong or Macao passport, she will need a valid Chinese passport and must apply for a visa from the Italian Consulate serving the area where you live (see "jurisdiction" under each entry).
This page, once you have answered a few simple questions, will give you a list of the forms and requirements for a visa application. You should also check the website of the Italian Consulate serving your area, as different consulates may have slightly different requirements. The Consulate's website will tell you how to make an appointment.
According to USA Today:
If you stay in the U.S. on the basis of a United States Permanent Resident Card -- or USCIS Form I-551, popularly known as the Green Card -- your ability to travel to Europe without a visa will depend on the country that has issued your passport. Your permanent residency status in the U.S. doesn't influence your need for visas, while your actual citizenship does.
And according to this map (I assume you can locate Italy on a world map), your wife requires Italy visa prior to visit Italy and the EU.
Depends on where you live, find an Italian consulate/consulate general/embassy and ask them whether they handle visa affairs.