Your best proof that you are either a U.S. permanent resident or a conditional permanent resident (who received status either as an investor or as the spouse of a U.S. citizen whose marriage was less than two years old at the time of approval) is your permanent resident card, also known as a green card. Usually, you will receive the actual card in the mail within a few weeks of your application being approved or your entry to the United States.
If you enter the U.S. after being approved at a consulate, you will also get a stamp in your passport (an "I-551") when you enter. This stamp serves as temporary evidence of your permanent residence while you are waiting for your green card to be mailed to you. You can show this stamp to employers or use it to travel in and out of the United States.
If you get permanent residence by adjusting your status (applying entirely within the U.S.), most officers of U.S. Citizenship and Immigration Services (USCIS) are unwilling to put a stamp in your passport. If you know you will need proof of your permanent residence before you get your green card, however; for example, if you need to leave the U.S.; they might consent to giving you the temporary stamp.
After your application for a green card has been approved, but before you get the actual card in the mail, you will receive an Approval Notice and a Welcome Notice from USCIS. You will no doubt be relieved to see these notices, but do not try to use them as if they were your green card. If you leave the U.S., you cannot use them to get back in.
If you are over the age of 18, the law requires you to carry your green card or other evidence of your U.S. immigration status at all times. Keep a photocopy of the card in a safe place, however, in case it gets lost or stolen. Having a copy will make it much easier to get a replacement card from USCIS.
With your U.S. residency, you are eligible for a Social Security Number (SSN). The SSN is given to all people legally living and working in the United States, to identify them and allow them to pay into a system of retirement insurance.
You might have already applied for a Social Security number, for example if you received a work permit before getting your green card. If not, now is an excellent time to apply. You will need this number before you start working. Your new employer will ask for it in order to file taxes on your behalf.
To apply for an SSN, you'll need to visit your local Social Security Administration (SSA) office, which you can find via www.ssa.gov. This is required of non-citizens because you'll need to show original documents.
Unfortunately, all SSA offices are closed to outside visits as of early 2020, owing to the coronavirus pandemic. Even after they reopen, expect delays into the foreseeable future, unless new procedures allow digital applications.