Gambling winnings need to be reported on your federal return as "Other Income" on page 1 of the return and they will carryover to your resident state tax return. If the State where the winnings occurred withheld taxes you will file a Non-Resident Income tax return with that State. In TurboTax, you prepare the nonresident state return first and then the resident state return. This will keep you square with both states and allow you to avoid double-taxation at the state level.
When you file your resident return, it will ask you if you paid taxes to another state and you would say "yes" (assuming you file the nonresident return with the other state). Then your resident state will ask you how much you paid to the other state and you would enter the taxes paid to the other state from that state's return (not the withholding amount but the actual tax paid on the other state return). This way, your resident state will give you a credit for the taxes you paid to the other state so you are not taxed twice in each state.
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