If I moved to a state part way in the year, and all my income was made in another state, do I report the income I made while I lived in VT on VT state tax or do I put 0?

You will enter the income earned while a resident in VT in the VT part-year resident return even though earned from work in NY.  Also, I recommend completing your NY non-resident return first as the income earned in NY as a non-resident there and taxed by NY will be used to claim a credit on your resident VT return for taxes paid to another state. 

Your resident state typically taxes all income from all sources earned while a resident.  A state in which you worked but didn't live (non-resident) only taxes that state's source income.  That means the income is double taxed.  To mitigate that double taxation most resident states offer a credit for taxes paid to another state.