Generally, you do not need travel documents to travel within the continental U.S. Your military I.D. should be fine for flying, but don't travel outside the U.S. Be aware you should be notifying USCIS of an address change within 10 days of moving (you can do this online). You should consult with an experienced immigration attorney for help with the green card process.
The above is for the purpose of general information and is not meant to create an attorney-client relationship. Please seek the advise of an attorney on your individual circumstance.
A military ID with your picture on it is just fine to pass through the TSA controls at the airport. Make sure not to cross to Canada, though, even by "mistake"..
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
and a valid foreign passport, yes.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: email@example.com; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.