Your friend is correct. Go to the Social Security Office, fill out the form, meet with an official, and show the official your employment authorization card. You may need to wait 4 to 8 weeks until you get the Social Security card, but you may be able to call the Office to find out the number sooner. Make sure that you get some proof that you filed the application, if possible. At one time, it was easier to get the number sooner. If you live close to a major metropolitan city, then try to apply there, since the officials may be more experienced in what to do if they see the card.
This is general information, not legal advice, and does not create an attorney client relationship.