For a couple times during the past week, when I back up my vehicle immediately after I start the engine, I heard a loud screeching sound as I was backing up. But the sound does not come on if I wait a few minutes after I start the engine before I back up. Any thoughts?
Also, I recently moved to an apartment complex on top of a steep hill. I found that my vehicle shifts/jerks (once) going uphill. Any advice as to how I can manage the car better while going uphill? What could be the causes?
Make model year and miles on car please. This information will help.
I would suggest checking the transmission fluid level assuming it is an automatic. (You did not tell us that either) 
Sorry…I just realized I did not include those info. Those would have been helpful. 
It is an automatic transmission 1993 Chrysler Concorde. It’s got approx. 115,000 miles. Thanks again!
Also check your fanbelt (serpentine belt). This type of screeching is often the sign of a stretched/slipping fan/serpentine belt and/or a bad tensioner.
Do you get the screech when you start out immediately in forward? If so, it could be the s-belt. If not, I doubt it. By any chance are you cutting hard left or right as you’re backing up? The belt that drives the PS pump (likely the s-belt) could be slipping when cold and gets a little better grip when the engine is warmer.
On the jerky uphill shift, are you going at a speed that you would normally shift to a higher gear anyway? You didn’t say if it was upshifting or downshifting. If it’s upshifting, either slow down a bit or drop it into a lower gear before you start the hill climb.
Actually, because the reverse gear is so much higher ratio than any other the engine will be revving higher than it would going forwrad and the screeching can be noticed there first.
However, it was just an added suggestion to those already posted anyway. And they’ll usually screech in forward too…however I got the impression that he started out in reverse every morning.