I consider all travel that involves engaging with the landscape, culture and/or people to be a form of pilgrimage.
Some of the more consciously pilgrimish travel I have done, though, included going to Down House where Darwin lived and walking along the gravel path where he thought about evolution, and having a conversation about evolution. I think the re-enactment element was important there.
Another example was going to Canterbury Cathedral. I am not a Christian but I find the story of Thomas a Becket moving, and I like Chaucer's Canterbury Tales and Jean Anouilh's play Becket.
Visiting stone circles always feels like a pilgrimage to me. They are beautiful and numinous places, and some archaeologists think they were made to represent a microcosm of the landscape.
Landscape itself - the wild places - is a place of pilgrimage for me; it is where I go to feel renewed and refreshed.
I also think that places where other people have made a connection with the numinous are special. As T S Eliot wrote in Little Gidding,
If you came this way,
Taking any route, starting from anywhere,
At any time or at any season,
It would always be the same: you would have to put off
Sense and notion. You are not here to verify,
Instruct yourself, or inform curiosity
Or carry report. You are here to kneel
Where prayer has been valid. And prayer is more
Than an order of words, the conscious occupation
Of the praying mind, or the sound of the voice praying.
And what the dead had no speech for, when living,
They can tell you, being dead: the communication
Of the dead is tongued with fire beyond the language of the living.
Here, the intersection of the timeless moment
Is England and nowhere. Never and always.