What's your advice on moving for the first time?

I recently got a job after years of moving back home with my parents after college, and it's always been my dream to move out of the poor rural town I live into a major city, and find work that uses my degree. As someone who never rented an apartment before, I lived in the dorms on campus, how much money should I save up to follow my dreams? The sooner I move out the better, because I hate living in the small town I really don't fit in here.