My Toyota Avalong XL 2000 got a new timing belt.
With the original timing belt the normal idling speed was 1000 rnd/min, with the new one it dropped to approximately 750 rnd/min. Plus I feel some shaking. Plus I feel a few 'brr' sounds when I turn off the car. What was done wrong?

either they did not properly line up the gear teeth when installing the belt, or they installed something that they had to remove, incorrectly.

Thanks. Is that kind of work for a high level mechanics only?
I got my old belt, and the box from the new belt. That is a 'Conti Synchrobelt Kit'. The label on the box says 'TB257K1. Replaces:
Gates: TCK257
NAPA: 2522571
Goodyear: GTK0257'

Could you, please tell if the kit is right for Toyota Avalon 2000?
How to proove to them that something is wrong?

I am not sure what kit is the right one. I would bring your car back to where you got the work done, and tell them something isn't right.. They should fix what they did wrong.

750 rpm is a good idle speed. Whatever is wrong, it is not the idle speed.

I worked on an Avalon V6 (1995) that had one bad fuel injector. It was noticeable as soon as you put it in gear to drive around, and I could tell it was misifiring at an idle. But... the idle speed before the repair and after the repair was exactly the same. Just smoother with all cylinders working. I also replaced the timing belt and the two pulleys. That didnt change the idle speed at all. The computer was choosing the idle speed, not me. You might have a sticking throttle plate, or a poorly adjusted throttle cable, or a sticking idle air control valve. Those would affect the idle.