I have a 1992 Taurus

I have a 1992 Taurus. A few months back, while at work, I left my lights on and low and behold my battery died. I got a jump and was good for the next week or so. I am leaving work the following week and barely got the car started, get home and the following day the car wouldn't start at all. Wouldn't start with a jump either. I proceed to purchase a new battery, get it in the car and it still won't start, even again with a jump. Whenever I get in the car and everything's hooked up, I turn the key to the position where all the dash lights in the car come on and I hear a whirring noise that stops soon after. I turn the key further as if to actually start the car and nothing, doesn't start, no other noises. Any suggestions? My father suggested manually turning the engine or something due to the nature of the whirring sound and the teeth not grasping or missing teeth, but as I'm not too car savvy I'm not sure if this would help anything at all because the whirring sound isn't constant (if it's supposed to be?).