Transmission shifting issues

Okay, so I bought this car not long ago from a friend. It seemed to run fine. Then I took it up to my sister-on-law's house one day (which is up in some hills- compared to the flatter valley area where I live) and right before I reached her house I was stopped, getting ready to turn onto her road and when I hit the gas it was like the car didn't want to move. It very slowly did finally turn and as I was driving past a couple houses to get to her driveway, the car was getting hotter than usual so by the time I got there and shut it off it was in between the half-mark and the red zone on my thermostat. I let it cool down for a few hours. Then I tried to drive it again and this time it heated up really quick but didn't get too hot (it usually sits at halfway) but it felt like it wasn't shifting at all so I turned around and parked it again. We checked the fluids- there was an oil leak but everything else was not leaking and full with fluids. The next day I took it to the store a few times and it ran perfectly fine. Then later that evening I went to go home (about a 30 minute drive) and it was driving fine until about halfway home and then the speedometer started jumping up and down and then stopping and then jumping all the way back up and then bouncing some more. The car, at this time though was still running fine as far as I could tell and I made it home. The next day I tried to drive it to my mechanics house and it was doing the "wouldn't shift out of first" thing again. He changed the transmission speed sensor on it but that fixed nothing. Any ideas before I drive this darn car off a cliff?