Guys I just moved my stuff from NYC to Toronto couple of weeks back. Considering I have moved quite a few times in the past few years, here is my advice -
If it is a small studio or even 1 bedroom or two bedroom apartment, you are better off just selling big stuff and buying new stuff in Canada unless your current stuff is very expensive. I was living in a 2 bedroom apartment, I sold all big items except electronics, expensive kitchen items, Projector etc.
Packed everything remaining in boxes and put it at my sister's place. You can put it at a storage facility as well.
Then I packed my bags and moved to Canada. Declared things to follow.
After I bought my car, I drove back and picked up my remaining stuff and went back. My SUV was full of stuff but absolutely no issues at immigration. You can also rent a car and drive back to US to pick stuff.
But if you have a big house and you intend to move everything, you can rent a truck and use a service like "Hire a helper" to load stuff in the truck.
Cross the border, go through immigration and hire helpers to unload.
Please note in this case you will have to pay a considerable amount of money for truck rental because it will be one way rental with drop off charge which could be easily over 1K bucks. Also, the pricing of the trucks can be tricky, means they will charge you some fixed money and will also charge you per mile driven, which depending on your case add another big chunk of money. To it add cost of hiring people to load and unload which could be about another 300 to 400 bucks.
You can get estimates and see what is better money wise as well as hassle wise.