Q: Car is having issues starting

Saturday I had gone out to start my car to go to the store. At first it started and sounded different than usual, Saturday was the first cold/snowy day here in Indiana. As I was cleaning the little bit if snow off the car it day. I turn the ignition all the way off and waited a few seconds before trying to start it. I tried to turn it over a few time. It clicked and made a slow whining noise. After the third time it started and stayed on. I let it run and warm up before I left. Today I went to go look at new batteries because I thought that was the issue. After I got to the store to look I went out to the car and tried to start it. It wouldn't start or make any noise at all. I got somebody to jump it. We put the cables on and I waited a few minutes before turning it over. And again it wouldn't start. I tried a few more time over the next 15 minutes, still nothing. I put a new battery. So now it starts but is still making that slow, whining noise. It barely starts.

My car has 265000 miles.
My car has an automatic transmission.