You say your girlfriend is a resident of Canada. Do you mean to say that she is a permanent resident? If she is, she can sponsor you for PR.
You can not officially move to Canada until you get your PR so you might not want to show up at the border with all your stuff in a u-haul.
If you apply outland, you may visit your wife for up to 6 months at a time. If you want to apply to extend your visitor status, you can do that as long as you prove when you entered the country. If you are a US citizen, they may not stamp your passport so you should either ask them to do that or ask for a visitor record. A visitor record is when they put your visa in writing, when you entered, how long you can stay. If you explain that your wife will sponsor you, they may decide to give you a year instead of 6 months.
If you want to apply inland, that is also a choice. In that case, you will have to stay in Canada during the processing because it is a requirement of inland that you are residing in Canada.