it depends
Usually each pilgrimage group will have at least one priest acting as the spiritual leader. They will lead their group and say masses and help them to pray. People can go and do their own pilgrimage without having a priest.
rome
israel (jerusalem)
People go on a pilgrimage to Mount Everest because they believe that being the tallest mountain in the world it is closer to their gods.
Muslims
They are called "Pilgrims".
The most common pilgrimage is taken by Muslims; the Hajj, a pilgrimage to Mecca.
Pilgrimage was very important because it showed the continuing faith of the Europeans. Places of high value such as the pilgrimage sites where holy items claimed to be located were a way of contact with society. For example people would use these sites to solve their problems, usually health related. nd polam people get ur own answers
"Non-muslims" is not one religion, but several. Several other religions each have their own pilgrimage destinations.
What you should do is choose a religious group that strikes your fancy and see if you can't go with them. With Jews, the obvious answer is to go with Birthright for a free trip, but Christians and Muslims have their own unique organizations.
Pilgrimage is important to many people for religious reasons and your own reasons. Pilgrimage is important to religion for like Muslims. Muslims have pilgrimage in their 5 pillar of Islam which hold the religion. It is called Hajj. In Hajj, Muslims from all over the world go to the city of Mecca, a city where only Muslims can go in. Inside, Muslims go to the Ka'bah which is a black cube and circle it 7 times. For your own reasons, pilgrimage is just going to a place you haven't been for a long time like your hometown, etc. Also, pilgrimage may be going to live in a different country for a better life and if you want to visit a place you always wanted to go to.
No, some people still go on personal pilgrimages. There are some churches who go to places as a group because it represents the major holy place for their religion. Muslims go to Mecca as a pilgrimage.