Here are 3 Bible verses about going on pilgrimage, curated from both the Old and New Testaments of the Holy Bible, King James Version, arranged from most to least relevant.
John 4:20 KJV
20 Our fathers worshipped in this mountain; and ye say, that in Jerusalem is the place where men ought to worship.
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Acts 17:24 KJV
24 God that made the world and all things therein, seeing that he is Lord of heaven and earth, dwelleth not in temples made with hands;
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John 3:5 KJV
5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.