Yes, your rental income from your house in state A must be reported on your Federal return. In general, it will be reported on Schedule E Rental Income and Expenses unless you are not renting it at a fair market rental price. If that is the case, your rental income is still reported, but your rental expenses for that property will be limited.
Because you moved during the year, you will be filing two part-year resident state returns for state A and state B. When you allocate your income to each state return based on when it was earned, the rental income (or loss) will be allocated to state A because that is where the property is located.
As for claiming rental expenses incurred in state B, there is not a Federal tax deduction applicable for this if you are referring to your having to rent a place to live due to a job change.
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