How should I get a job in finance?

I'm a psychology/biology undergraduate senior graduating in the spring, help me get into a finance job in new york. After majoring in psychology and biology I've realized I don't want to go into either of those fields. I want to do something in finance, perhaps corporate finance or investment banking instead. I don't have any finance experience directly, but I've held a bunch of summer/during the school year job/internships mostly dealing with medical research, or psychology industry jobs. So here's the question. Am I wasting my time trying to directly apply for an entry level job in some of the big banks since I don't have any specific industry experience? Should I try to find my way in through connections? Also, what kind of starting salary could I expect. Some more background, I'm graduating with approx 3.8 average from a top 50 school in the new york area. Like I said I've held some jobs and internships, but nothing crazy prestigious or anything. I'd appreciate any advice the mefiers have to offer.