Layover in Rome, What to do?

Hello, I have a layover in Rome (coming from Amsterdam to my home country). The layover is from 11:10 AM (landing) to 9:50 PM (takeoff). First, I never had a layover in a flight, and want to check what is the routine? The luggage will be kept with the airline? After going down from the plane, should I directly... show more Hello, I have a layover in Rome (coming from Amsterdam to my home country). The layover is from 11:10 AM (landing) to 9:50 PM (takeoff).

First, I never had a layover in a flight, and want to check what is the routine? The luggage will be kept with the airline? After going down from the plane, should I directly proceed to checkout? what are the steps to leave the airport and come back again?

Second, how to go to the Vatican the fastest way from FCO Airport? Note that it will be on Easter Monday 2018. So I've read that by that time Vatican/Rome will be crowded.

Third, I want to check the Vatican and Rome (Trevi fountain, Panteon, etc..). So what's best to do? Go to Rome first then to the Vatican? or Vatican first?

Last point, at what time should I go back to the airport ? Knowing the the plane leaves at 9:50 PM.

Thanks!
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