2000 Saturn SC1 Problems with Shifting Unusual Noise

I really don't think it is inside the transmission. Do this. I want you to find out the engine speed for your shift points. Meaning, when you shift from second to third gear, look at your engine speed meter or tachometer. Example you shift at 4000 rpm.

Now you need to simulate that with the car in park, step on the gas to 4000 rpm (or whatever your shift point is) and list to the noise. If the noise is present, then it is not in your transmission. It is a lot easier with an assistant stepping on the gas, so you can look and listen for the noise from under the hood.

I suspect your automatic belt tensioner.