How does my tax return change if I lived in a state that didnt have a state income taxes for half the year then moved to a state that does have state income taxes?

[Critter]

Simple ... for the state that does have an income tax you will file a part year return and get taxed on only what you earned while a resident of that state. 

[rjs]

Your federal tax return is not affected unless you had state income tax withheld in your new state and you itemized deductions on your federal return. The state tax that you had withheld during the year could increase your itemized deduction for state and local taxes, if you have not hit the $10,000 cap.

[TomD8]

There is one additional wrinkle.  If you were working in the state with the income tax prior to moving there, that state could tax you as a non-resident on that portion of your income.