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For your first year, you'll probably have to file in both countries as you'll likely have earned income in both Canada and the US. The classification of which country you're a resident is based on which country you've spent 183 days in. (that's basically half the year)
You'll definitely need your visa first before you can do anything else like get a Social Security number (so you can get paid or open a bank account).
If you're paying for this out of pocket, then I'd suggest starting with trying to get a TN visa. (I'm assuming you're going for some sort of professional job) It's cheap (around $50 when I did it) and doesn't require a whole lot of work to get. You can probably do it without any immigration lawyers as long as you meet the requirements. Once you're settled in and have an employer, you can then have them sponsor you for a visa or green card.
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