How do you replace rear differential ring gear and pinion shaft 1980 fiat 124?

You will want to grab a factory service manual or a Haynes manual. Haynes is available new, the factory manual may be available electronically, or via ebay or Fiat sites. The general process is straightforward and includes removing the wheels, brake calipers, brake caliper mounting brackets, rotors, and axle shafts. Remove the nut on the front of the pinion shaft, pull the drive shaft yoke off of the front of the pinion, remove the 8 bolts the retain the rear cover, remove the four bolts that retain the two differential bearing caps, remove the caps. Note any shims used to position the differential carrier. Lift out the differential carrier to reveal the pinion shaft. The ring gear just bolts onto the differential carrier, so replacement is simple enough. The pinion likely must be pressed out from the front toward the back. New shims may be needed, so purchase these ahead of time. Also check and consider replacing the bearings, as well as the front seal and the back cover plate seal. This write up is general / typical, but you will want to follow the service manual instructions.