Automatics are much more complicated beasts. There are some things you can check, but a qualified Professional should really handle this. Make sure the transmission fluid is sitting at the appropriate level. Make sure the fluid is clean. Make sure there is no "burnt" smell from the fluid. You can try replacing the filter. A shift modulator/solenoid may be failing. ...possibly torque converter?
The grinding suggests the gears are in motion when they shouldn't be and/or are not aligned/synchronized properly.
I see these as "internal" problems.
Low fluid and/or a low-flowing filter will usually manifest as "not going into gear right away" and "slipping". Burnt clutch bands (yes automatics have clutches) usually results in slipping in gear too.
Honestly though, if anything, it sounds like there is "too much pressure" in the system. I would suspect the system has been overfilled. ...that could potentially cause grinding, but I would expect it to be heard in other gears too, but maybe once it grinds, the shaft is synchronized for the shift into first/drive.
Have you had "hard" shifts in the past? Does it grind if you go into 1st first instead of reverse?
You can take the car to AAMCO or someone and just request a "service". Probably around $100 at this point in history, but that will get you a fluid and filter change, knowledge that the fluid level is correct, and most importantly, qualified eyes on the system.
Start with the procedure for checking the fluid level.
I know very little about automatics, and these 4l60 transmissions apparently have some quirks. I bought a '96 Corvette a couple months ago which is basically a 4th generation Formula/Trans Am in refined 3rd generation Firebird clothing. It's the first automatic I've owned in over 35 years. When my Ignition Control Module failed days after purchase, I shifted into Neutral to try restarting while in motion. When the shifter moved from D to N, I heard a very light grinding noise, sounded like plastic gears. Did it again later when the engine stalled again. It does NOT make this noise when the engine is running and the rear wheels are in motion, doesn't make it sitting still. ...but engine at 0RPM, transmission "turning", there is a grinding shifting out of gear. Couldn't find anything on this, transmission seems fine, but I can easily replicate it anytime / anywhere.
Also, and this might have some relation, when I start the 'vette on a level surface like my carport, it lunges forward! This is considered "normal" and "acceptable" behavior by GM. I think there was a TSB about it, but there is no fix as it's not considered a problem. ...but, it does suggest that even with a "perfect" transmission something isn't "releasing" as well as it should and the clutches are engaged a little. I have the 4l60e (E for "electronic"). I do not know if this behavior happens with the older 4l60, or which variation came in your 'bird.
Have you tried starting the engine in Neutral and shifting into D or R from there to see if there is any change to symptoms/conditions?
Does the symptom change between a cold and hot transmission?
...anyway, go check the fluid level/color/smell and report back.
