Q: 1994 Ford Taurus will not start after battery/starter replacement

Bought this car a few weeks ago and after a few days, it wouldn't start. I thought I maybe left a light on and after 20 minutes of trying, my friend successfully jumped it and I drove it home. It wouldn't start again the next morning so I figured it needed a new battery as this same thing had happened to me a couple years ago with a different car. After an hour and a half of installing the battery and cleaning the bad corrosion, it started right up. I drove it around a while just to make sure it would run. It wouldn't start again the next day. At this point we started thinking it could be the starter as it would click when we turned the key. We replaced it and thought we had figured out the problem because it started again, but after we turned it off, it wouldn't start again. It did this a few more times where it would start but after shut off, wouldn't start up again. It's clear the car is getting power, but it seems to be picky about when it wants to start. Whats our next move?
My car has an automatic transmission.