State tax filing

@Mac1287  You are mostly correct.  Hopefully I can clear up the guidance for you.  Here's what you'll need to do:

New York.  You file a part-year return.  However, the program interview will ask you about New York income you earned during your nonresidency period.  This will be the double-taxed income for your New Jersey return.  

Later on, you are interviewed on your NYC residency.  Make sure you mark your NYC residency dates also.  This should allow the program to prorate your NYC-taxable income as well, because you are correct that NYC only taxes residents of NYC.  Your refund for New York should increase by the amount of NYC tax withheld on income that should not be taxable there.

New Jersey.  First make sure you mark the New York income as not taxed in New Jersey when you get to the screen About this income.  You will get this screen if you have amounts listed on your W2 for both NJ and NY, because New Jersey requires both lines to report on the return, causing a double-up of income.  Then, when you get to the Income taxed by another state screen, you will calculate out the New York tax on the double-taxed income as you suggest (NJ gross income divided by NY gross income, multiplied by NY tax).  

You do not need to worry about tax brackets when it comes to this credit.  That is already factored into the equation.  New Jersey's credit is the lower of what New York taxes the double-taxed income, or what New Jersey taxes the double-taxed income.  And the New York income is actually subject to New Jersey tax brackets.  What New Jersey does (and many other states, including New York) is pretend that all income is taxable in New Jersey, so they can apply the tax brackets.  Then the figure is prorated to the amount of income earned in New Jersey.  But since New York's tax rate is still higher than New Jersey's (in general), it is likely that the end result is $0 tax for New Jersey.  

Feel free to ask any additional clarifying questions.