it's really relatively easy to do..although i'm not sure if it would be possible by your second quarter, as david said, it would depend on your individual situation.
basically though, you need to show that your income exceeds your expenses for a period of a year prior to the quarter you want to get residency for and that you have been been independently paying for it. for example, if you are currently attending undergrad at OSU, your tuition would be part of your expenses and that either has to have been paid by loans, money you're currently earning, or savings (paying with savings is a little trickier because you have to show that the money was there for a year prior to the 'review year'...meaning 2 years prior to the quarter for which you want residency).
so if your parents or other family have contributed towards your tuition THIS YEAR, then you pretty much have to wait until next fall to get residency.
certain things like a cell phone, your parents can still technically pay for, but you have to show that you COULD pay for it if you needed to, meaning you still include that amount in your list of expenses.
this probably sounds slightly more confusing than it actually is. once you pick up the residency form/application and actually see it, it all makes more sense.
just to wrap things up a bit, some of the things you DEFINITELY should do, and as soon as possible are get an Ohio driver's license, register to vote in Ohio, if you have a car, change the title over to an Ohio title and get Ohio plates.