I just moved to New York from Toronto, Canada and I'm trying to understand how much I'm going to be taxed on my income. I'm single without any complicated investments. I'm a Canadian citizen in New York on a work visa (TN status). Would I still get taxed in Canada? And could I use the fact that I didn't get paid in the USA for the first 9 months to get a tax return?
This question would be better asked before moving, not after.
The Canada Revenue Agency follow a set of rules when they determine if you are a Canadian resident for tax purposes; based on the limited information you provided, you're likely to be subject to Canadian taxes.
As for the US tax return, it is just that -- a return of some of the taxes you already paid back to you. It will be based on what you will have paid in US taxes by the end of the year and your actual US tax obligations at that time.
Your best bet would be to talk to an accountant familiar with cross-border taxation.