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For domestic flights within the U.S., you will go through exactly the same processes as U.S. citizens, which are very similar to the processes for airline travelers around the world. You will have to show a government-issued photo ID (typically, a passport for foreigners) to pass through security screenings.
Note that many U.S. carriers charge a fee for checked baggage on domestic flights that are booked on separate tickets than the international flights, unless one has elite status in a frequent-flyer program, or is flying in a premium cabin. If you will be flying on either JetBlue or Southwest, you will get a free checked-baggage allowance for a limited number of bags.