saturn 2001 sw1 car jerks when shifting into gears,

Not quite sure of you description...possibly worn engine mounts? If you put it in low gear and get up to about 20MPH and then tap on and off the throttle , do you get that same 'hard shift' feel?

I've had this happen, if the idle is correct and if the shit from 1-2 or 2-3 is hard then the trans shift solenoid is sticking and will probably need replacing for around $250. The bad motor mount will also transfer all of the engine vibration/shock to the body and is easily checked with a screwdriver or small prybar, try to move the engine side of the mount from front to back while watching the rubber mount material. It should be solid and not separate. A mount is under $30 and can be replaced by the average person with caution. Good luck!