You can not officially move to Canada until you have PR (= permanent residency which is similar to your green card in the US). It is therefore not advisable to show up with a U-haul full of your stuff. However, you are allowed to visit, so showing up with a suitcase would be fine.
Once you are living together, you can see how it goes. If you want to get married after a couple of months, you can do that and he could then apply to sponsor you right away. Otherwise, you need to extend your visitor status after 6 months with a reason such as "exploring relationship" and then you can stay another 6 months at the end of you would have lived together for 12 months and would in Canada be classified as common-law partners and then he could also apply to sponsor you, as long as you have kept proof of your living together during those 12 months.
The downside to living in Canada on visitor status is that you would not have access to health care and you can't work or study. You would be allowed to take short courses though, under 6 months.
If you decided to go the other way, that is bring him to the US, he would have to apply for a fiance visa to US immigration and then you would have 90 days to get married after he arrives or he'd have to leave again.
