These are notes for myself because it's that time of year again and I keep forgetting. I am not a tax lawyer and this is not tax advice. Follow at your own risk, I am not responsible for anything. This is for a Canadian citizen on an F-1 visa doing a PhD in the US.
On the US Side
I recommend doing this side first.
Income sources
In general you only pay taxes on US-sourced income. If you are getting fellowship money from NSERC this does not need to be reported at all as it is Canada-sourced income.
Fellowship income is taxable at the federal level and is automatically withheld at a 14% rate. It is not taxable at the state level in New Jersey. This will come as a 1042-S.
Teaching/assistantship/research income is taxable at the federal and is automatically withheld at a 14% rate. It is also taxable at the state level, and will be withheld at some rate which you can find in this table [3]. This will come as a W-2.
Interest received from a US bank (e.g., in a HYSA) is not taxable at the federal level for NRAs [1]. It does however need to be reported. It is also not taxable at the state level in New Jersey, as it counts as income from intangibles, which is typically sourced to the state of residence of the recipient (which in this case is Canada), and New Jersey only taxes you on income sourced to New Jersey [2]. Again it does need to be reported, and in the NJ-1040NR it will go in 16A, but you will put $0 for 16B. This will come as a 1099-INT, and if you have filled out the W-8BEN form correctly, the bank will not withhold any taxes on it.
Dividends from investments in a non-tax-advantaged brokerage account are taxable at the federal level at a 15% rate, which is due to a US/Canada treaty (normally this would be 30%). Your dividends will typically be separated into "ordinary" (normally taxed at 30%, for Canadians this is 15%) and "qualified" dividends (taxed at the more favorable long-term capital gains rates of either 0%, 15%, or 20% depending on your income), but as an NRA you will not be able to take advantage of the lower tax rates for qualified dividends and all dividends will be taxed at the flat ordinary dividends rate. At the state level you can also treat them as intangible income and and the process is the same as for bank interest: put the appropriate amount in 17A, and $0 in 17B. This will come as a 1099-DIV, and again there should be no withholding from your brokerage if you have filled out the W-8BEN form.
If you start work with a new employer you should fill out a W-4 form. You should also let them know that you are exempt from FICA taxes under IRC Section 3121(b)(19) and you will probably have to send them your I-20. If you open a bank account or a brokerage account you should fill out a W-8BEN form.
Filing
For your federal taxes fill out 1040NR and attach all of the above forms (W2, 1042-S, 1099-INT, 1099-DIV). There are two other things that you will need to attach. The first is schedule OI (other information), which says "I am a Canadian resident" and provides other miscellaneous information. The second is attach form 8843, which is just a form that says "I am an F-1 student and I am exempt from the substantial presence test". I am using Sprintax to do this and from my experience it seems to be pretty good at getting things right.
For your state taxes fill out NJ-1040NR, and attach the W-2 and 1042-S. You do not need to attach the 1099-INT or 1099-DIV unless there is New Jersey withholding on them (which there shouldn't be). I think Sprintax doesn't do the 1099 income properly and it is fairly simple to do the NJ-1040NR by hand.
On the Canadian side
I recommend doing this side second.
You probably count as a factual resident of Canada while you are not a tax resident of the US.
At the beginning of each year ask the university for a TL11A. You can use this to gain an absurb amount of educational tax credits (although they are not as good as one might think at first glance, more on that later) especially if you are "paying" PhD tuition to the tune of $70k USD each year. This is valued at 15% (i.e., $100 of tuition paid = $15 of tax credits). This also very nicely rolls over from year to year if you can't use it all in one year so you can accumulate a lot over time, and you can also transfer a small amount to family members each year. You do not need to attach this to your tax return, but the CRA will probably ask for it if you are claiming that much so hold on to it.
In general the guidelines are that you will pay the Canadian government whatever you would have paid if you were working in Canada, minus the amount you already paid the US government. So if your US taxes are higher, you pay nothing to Canada. But if your Canadian taxes would have been higher, you just pay the difference to Canada. This is meant to prevent double taxation, and is done through foreign tax credits, which you can claim on your Canadian taxes by reporting the amount you paid to the US government. Very unfortunately, the foreign tax credit does not roll over, and is also of lower priority than the educational tax credits. That is, educational tax credits will be applied first until either you have no taxes left to pay, or you have used up all of your educational tax credits, and only then will the foreign tax credit be applied. This means that if you have a lot of educational tax credits, you may not be able to use the foreign tax credit at all, and you will end up effectively paying taxes to both the US and Canadian governments on the same income. Unfortunately there doesn't seem to be much that you can do about it, and it seems to be a fairly well known problem [4, 5, 6, 8].
I use Wealthsimple Tax and it is easy enough. Report Canadian income as regular income, and US income as foreign income. Your Canadian fellowship income will come as a T4A and should be tax-exempt [7]. Here, "US income" is just the amount on your W-2. Do not include US-sourced fellowship income (1042-S), as it should also be tax-exempt [7]. However, if you paid federal taxes to the IRS on your US-sourced fellowship income, do put that down as foreign taxes paid. This can lead to the weird situation where you report paying more tax than you had income [8].