How are different parts of airliners handled after the end of life-cycle?
Various parts of the airliners face different fates once it has been decided to breakup the aircraft. Most of the parts are reused in one way or the another. The dismantling of the aircraft starts after removal of hazardous liquids and radioactive components.


Engines- Usually, the aircraft and engines will be at different stages of their lives. The engines are removed and sold for operating in other aircraft. In case the engine life is also considered over, either its components will be used for spare parts or it too will be scrapped for materials.




Source: jalopnik.com


Avionics- The avionics components are usually valuable and are removed for use in other aircraft. Since they are line replaceable units anyway, this should pose no problem.




Stripped down cockpit of 747-400; Image from aviationweek.com


Interior- The interior of the aircraft, like the seats, entertainment systems are stripped and sold off. Items in the galley (like coffee makers), the air-conditioning systems also face the same fate.




Stripped cabin of 747-400; Image from aviationweek.com


Some items like the APU, actuators, landing gear, pipings, etc, if they have life are removed and sold for use in other aircraft (after appropriate checks and certification).
In case the aircraft has removable winglets, they are removed as well, for re-use.
Any hazardous materials like composites, depleted Uranium (used as counterweights in 747s, 727s etc.) are removed and disposed off.
The aircraft (structure) is then cut to pieces and metal parts are melted and reforged for reuse in industries, with unusable parts finding its way to landfill.




Source: tarmacaerosave.aero

The following figure from ICAO report Aircraft End-of-Life: Scrapping and Recycling shows how the materials are reused.



Image from What's new in aircraft materials recycling? ICAO workshop presentation

In general, stripping down and recycling of aircraft involves millions of dollars and is highly efficient, with over 90% reuse of materials being quoted. It remains to bee seen what happens to material reclamation as more and more aircraft components are made from composites.