To travel, you usually need your permanent resident card, a valid passport, and whatever visas are required by the country you intend to visit. While the US does not require permanent residents to have a valid passport to re-enter the US, foreign countries and airlines require you to have a passport.
As the list indicates, the TSA accepts a variety of forms of documentation for domestic flights. Notably for many foreign nationals, permanent resident cards and employment authorization cards are both acceptable forms of documentation for domestic flights.
ESTA is an automated system that determines the eligibility of visitors to travel to the United States under the Visa Waiver Program (VWP). Authorization via ESTA does not determine whether a traveler is admissible to the United States.
If you have a green card, it's not necessary to carry a passport with you for domestic travel within the United States. That said, it's never a bad idea to have plenty of documentation, and if you have a passport, you may wish to keep that with you throughout your domestic travel as well.
If you are a permanent or conditional permanent resident who has been outside the U.S. for one year or longer, apply for a re-entry permit before you travel. Use Form I-131 - Application for Travel Document. For permanent residents, the re-entry permit is valid for two years from the date of issue.
Form I-131, Application for Travel Documents, Parole Documents, and Arrival/Departure Records, is used to apply for the following documents: Reentry Permits, Refugee Travel Documents, Temporary Protected Status (TPS) Travel Authorization Documents, Advance Parole Documents, initial Parole Documents for noncitizens ...
Canadian permanent residents may need a non-immigrant visa to enter the United States. You must obtain this visa from the U.S. authorities before entering the country. You must also have a valid passport from your country of citizenship.
Provide proof of lawful permanent residence such as your Green Card (Form I-151, I-551), Re-entry Permit/Travel Document, etc. Tell us your dates of travel outside of the U.S. (airline tickets, boarding passes, passport stamps, etc.)