I'm not really sure what's going on with my engine

My battery died a few weeks ago because of extreme corrosion. My step dad cleaned it but it still wouldn't start so he changed the battery. The battery he put in is slightly smaller but he said that the other car it was in was the same size engine so that it should run fine. The next morning when I went to leave for work I cranked up and then immediately died again. So he came home and got it to start by pushing down on the gas peddle as he cranked the key. After a few days of this it stopped needing extra help to get started so I let it be. Then about a week later it started happening again. For the past week I've had to push the gas peddle while cranking the key and then rush to put it in gear otherwise it fades out and dies. Once I get going down the road everything is okay. However when I pause upward on a hill my car dies again and sometimes if I sit still for a long period of time like when I'm in traffic it will die too. This happens even with a full tank of gas. And I'm starting to wonder if it's that the battery is too small. I've considered just trading it in but I really don't have the money to make car payments so I want to try my best to figure out on my own what the problem is before taking it somewhere outrageously expensive.