Moving to SF from the East Cost (NJ)

I’m currently a college student in Jersey City, NJ and I’m planning on moving to San Francisco with my girlfriend. She already lives and goes to school there, but we’re not sure exactly how much we’d need to be able to make it there living on our own for a few years till we graduate.

We’re planning on moving together and a studio is fine with us. We’re also planning on having some money saved up before the move (5–6k) to help facilitate the move as much as possible. We’re just not sure if this would even be possible, or if we should start looking at other cities.

Could we do this on $2200—$2400 a month between both of us? Assuming I can find a job, which I’ll be looking for in advance.

The other city we were looking at was San Diego. If you guys have any information or experience that would be helpful for us we would be extremely happy to hear it. Thank you in advance!