Birth Certificate correction?

I have typos on my birth certificate. I was born in New york city. My moms last name along with mines. I was thinking of getting a pport but i'm being told it would be better to get my papers fixed so it can be easier. what should i do? My moms planning to cross into the U.S, and we live in Canada. My mom keeps telling me if i cross with her i might not be allowed back into Canada. I want to go so atleast if the amendment doesn't work out that i can get my license and all in my country. Should i go with her or should i stay?