That's good advice from rpenmanparker. It is hard to tell if cog or chainring teeth are worn, though, since they get cosmetic wear marks immediately and have odd shaped teeth to improve shifting. I would need to ride it and see if the chain jumps.

The right shifter should click into gears. (The lever under the brake goes bigger cog, the thumb shifter smaller cog.) The shift mechanism can get worn after many miles, and the shifts get "soft", "mushy", and vague. It's repairable, but that depends on the model of the shifter. Check the left side too.

My rear Veloce derailleur got worn pivots after 15000 miles. There was slop in the pivots, and I could wiggle the derailleur and feel movement. I think this would be less likely on a Record derailleur. The cheaper Veloce had small pressed-in pivot pins, and they redesigned it later for larger pivot pins, set farther apart.
So, just wiggle the derailleur body and see that it's tight. I could hold my worn derailleur by the frame mounting bolt area and the arm at the bottom pulley, and wiggle the lower pulley arm inward-outward a mm or more. Good derailleurs won't have any play.

Before installing your new purchase:
check the right shifter cable. Click into the correct gear that allows the head of the cable to be pushed out of the shifter. Push out a few inches. Cables fray and break where they make a sharp turn within the shifter body. When it starts fraying, you start having noisy, sloppy shifts or having to shift twice get to the next cog, then very soon, the cable breaks.

Cable sets

I always used real Campagnolo cable sets for cable & housing maintenance. It's expensive, though.