You do not have to say anything or sign anything.
Not exactly true. If you want to exercise your Fifth Amendment rights to remain silent, you must explicitly and clearly invoke them. “I am exercising my Fifth Amendment rights” works. “I want to speak to a lawyer” also works. Simply remaining silent can be interpreted by police as “being uncooperative” or even “being threatening”, and can bring associated consequences.
You have the right to make a local phone call, and if you’re calling your lawyer, police are not allowed to listen.
While this is true, you don’t get to choose where or more importantly when that phone call happens. You also won’t have your phone, and police are not obligated to let you get phone numbers from it.
Don’t argue, resist or obstruct the police, even if you believe your rights are being violated.
Unless you have reason to believe that your life is in danger because of the police. If you have reason to believe that you will be rendered to a foreign prison, for example, fight for your fucking life.
Finally, yes, you have these rights. They can and will be trampled upon. If you desire recourse, you’ll need to find and pay a lawyer, and spend lots of time navigating the justice system. The odds that you will succeed in that endeavor are vanishingly small. If you do eventually manage to get a judgment in your favor, good luck collecting on it.
Some rules vary by state. In VA if the police ask for your personal information “You can be required to give your name, ID, and other identifying information, such as your social security number. Do not lie, as lying about certain identifying information to police is a crime.”