First, congratulations on getting into your Grad School program! That's a great thing.
You are right about the price differences between Berkeley/Oakland and San Francisco. They can be very different but that is not say that you won't be able to find something in SF with your current budget. Because your question doesn't state your budget I'll just stick to basics.
I would first suggest using Craig's List. That is our go-to resource for just about everything in SF and rentals are definitely no exception. If that doesn't work, there are plenty of rental/leasing agents or companies that can assist you. I have a rental division at my firm but they typically lean toward the luxury sector. I'm happy to refer you to some others if you'd like.
Wherever you end up, I wish you the best of luck in your upcoming program!