Hi Ocanchumani
The answer will depend on whether you had income in Utah before you moved to New York. If so, that income will be taxable in Utah but not New York. If you had investment income that continues, such as interest income, that will be taxable in New York after the date of the move. For example, if you earned $240 in interest income and moved July 1, $120 would be taxable in Utah and $120 in New York. This will be handled on the allocation schedules for each state.
You will probably need to file as a part year resident in Utah and a part year resident in New York. After the year of the move, you would only file in New York as a full year resident.
There is a change of address form For the IRS - Form 8822. If the IRS owes you a refund, it is important to get this to them if the refund might be mailed rather than direct deposited.
I hope I have understood your questions and answered them clearly.
Maxthetax