what kind of derailuer would go best with an early 2000s TREK 820. I'm rebuilding an old rusty one and the front and back are no good.?

It's fairly hard to kill off derailers entirely through simple neglect. Why not start with a good scrub and lube, and work the motion by hand first?

Treck 820 is listed as an entry-level bike, so I'd suggest Shimano Acera otherwise.

You need to figure out if your bike has the derailer hanger as a part of the frame, or as a part of the derailer, and then buy a derailer with the correct configuration.

Front. you need to match clamp size to seat tube diameter, and to get the pull direction right. MTBs are usually Top-Pull. I'm assuming you have a triple crank, so the derailer has to be able to cope with that too.

The 820 was typically equipped with whatever they had laying around that was cheap. Usually Shimano Tourney or Altus. Definitely nothing higher.

Now is the time to upgrade to an Acera derailleur. They are only a few dollars more then what is on the bike.

go to a bike shop

they might even give the derailleurs to you

many of them would work

wle