The transmission makes it go, if it is stalling in park or neutral... there more than tranny problems.
I second the crank sensor first. Possibly MAF &/or Idle Air control?
I would get a second opinion from somewhere reputable. If it stalled everytime you put it in reverse or drive, then there would be a possible tranny problem, but If it won't stay running in park then there is a problem with the motor. Could be fuel related, could be a sensor.
I think more fuel than sensor because of "Also, sometimes it won't turn over at all, and I have to step on the throttle for it to turn over". But I could be wrong. May consider changing the fuel filter if you haven't lately and having it checked to make sure you have proper fuel pressure.