To start NY has a Convenience of the Employer Rule. Basically, this rule says if you are working remotely for your convenience then you are considered to be working in NY. If you are working remotely for your employers convenience, then you would not be considered to have income from NY sources.
If you DO meet the criteria to not be considered to have income from NY sources, then you would file a NY return to claim a refund of your withholdings. You would enter your NY income as $0 and your withholdings as however much was withheld for NY.
If you do NOT meet the criteria to be considered working remotely for the convenience of the employer, then ALL of your income would be taxable to NY. In this situation, you will file your NY return FIRST and include ALL the income earned in NY. This will all be taxed by NY. But you will then receive a credit on your NJ and PA returns.
For the time you were a PA resident, all of the income you received will be taxable to PA. PA tax rate is a flat 3.07%. If you did have to pay taxes to NY, then you will walk through the steps to take the credit for the taxes paid to NY. Be aware NY tax rates are generally higher than PA flat tax rate, so your credit will be limited to the amount of tax PA would charge on the income. You will not get a credit that results in a refund of the taxes you paid to NY from PA.
You will do the same thing for NJ as you do for PA. You will also need to enter the dates in which you lived in both states. This will help with the allocation of income to the proper states.
NJ taxes paid to Other States
PA Taxes paid to Another State
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