I'm not a financial expert, but here is what I would do: First, get a better idea of where you are really starting. Get your credit reports from all three major credit reporting agencies from annualcreditreport.com. Make sure everything in those is accurate and contest anything that isn't. Also, get your credit score for free (I think there are a couple places that you can do this, but I have only used Quizzle). Don't be too alarmed by your credit score -- the amount and number of loans we take out for medical school drag it down too.
Also call the hospital(s) ASAP and figure out if you qualify for any discounts or charity care. Since you presumably don't have any income outside of loans, you may be all to get part of all of the bill written off. Even if you can't, there are often significant discounts available for things like paying cash or paying within a certain period of time. Make a doable payment plan with the hospital and make your payments on time. Do not let your bills go to collections!
Unless the hospital bills are reported to the credit bureaus (which doesn't usually happen unless they go to collections), the hospital bills should not affect your credit, so just stick to the payment plan for those. Any extra money can go toward your loans, savings, or whatever else you need it to. There is a lot of information on these boards about the best strategy for tackling loans (some people swear by trying to pay them off as fast as possible, others swear by IBR and hoping for PSLF in 10 years, etc.), so I won't go into that. However, I will urge you to think about your finances as you pick your location for residency. Residency salaries don't vary that much across the country, but the cost of living does. $50,000 a year will go a lot farther in, say, Nebraska than New York City.