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.
To apply for a re-entry permit, you must file Form I-131 (“Application for Travel Document”). This form collects details about your planned trip, your foreign travel since becoming a green card holder, and whether you've been properly filing U.S. tax returns.
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.