You owe state income tax for all your world-wide income to the state where you permanently live (your domicile). Assuming your move is permanent, and you changed domiciles, then income that was paid to you when you were a resident of state A is taxed by state A and income paid to you when you were a resident of state B is taxed by state B.
For your situation, you will create a part-year resident NY return and a part-year resident CT return, by giving Turbotax the date you move out of NY and into CT. Then, you will be asked to manually allocate all of your income to each state. For each item of income (wages, dividends, bank interest, 1099-R, lottery prize, etc.) Turbotax will ask how much was paid while you were living in each state. If you changed jobs when you moved, then presumably, 100% of income from one job belongs to NY and 100% of income from the other job belongs to CT. You will be asked to provide the same information for all your other types of income.
