Once you move to a new country (Canada or USA), how to choose a bank to open an account? What are the important factor if you are also looking for the mortgage etc. after few years?
why feel that this is decision that you are locked into.
I would spend a short amount of time finding a bank/credit union that has a branch near where you work/live, and then create and fund the account. In the US banks and credit unions offer deposit insurance, a federal program that protects your money in the bank. Any money you deposit in the basic checking and savings accounts will be protected. There are limits to how much you can deposit, but that is covered in other questions on this site.
The rest of the requirements: website, fees, different products; those can wait for you to have more time to evaluate. You can then ask your neighbors and coworkers what bank they use.
While you might think that having a long relationship with a bank would be a plus when applying for a mortgage: there is no way to know if in x years they would even be a good choice. They could be great now, but not be in the future.