1998 Ford Taurus idle

I would start by checking the system for codes. Your description of the symptoms points toward a throttle position sensor. The computer uses this sensor to indicate that the throttle is closed while starting. With a bad sensor, the computer sees the wrong signal until you touch the throttle. While driving, this sensor is used to determine engine loading. If the computer sees a bad signal, it will default to a safe setting to prevent transmission damage. This setting raises the hydraulic pressures in the transmission and causes hard shifting.
The same bad signal from this sensor can cause both of the symptoms that you describe. If you do not have access to a scanner, you can go to many parts stores and they will scan your codes for free.