I've enjoyed limited travelling; but, have found myself drawn to natural wonders, museums & scientific exhibitions, and, particularly, architecture ... marvelling at the science & technology, engineering & aesthetics. From excavations of a recently found Roman arena/colleseum, to cathedrals & mosques, to Gaudi & Van der Rohe (Spain); Dutch & English colonial versus ancient ruins in Singapore & Jakarta; parliament houses & castles in England; and Gehry in downtown LA (near prehistoric tar pits). I also marvelled at the technology on show in glitz that is Las Vegas. The Bodies Exhibition at Luxor ... that's medical science worth seeing!

Religious influences & commissioning, seems to predominate in architecture; but public works infrastructure, law courts & libraries etc offer some relief from that.

I saw a dinosaur step out from under a waterfall and go foraging with what appeared to be an entire family of strange creatures at a fish restaurant for a meal on Friday night, that must count.

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Yeah. For sure. I’d love to see a volcano, the northern light, even a tornado or earthquake safely. It really has nothing to do with science though.

"Worship Science," No not actually worship it but TRUST in it, rely upon it, understand as well as make good use of it, then that would be a very BIG Yes.
As to "going on a scientific pilgramage" WHY on Earth would one need to do so when absolute EVERYWHERE around you there is the Wonders of Science and Knowledge to see, feel, hear, etc, etc.
"Look at a tree, look at a rock, look at yourself and what do you see, Science at it best, perhaps not its most PREFECT, BUT at it best at the time, you are, we are Science, the Science of the Universe itself, what more could you want." -William Anthony, 2018.

I do so regularly

One that stands out is the tour of the original Edison Factory(s....a complex) in Orange NJ....National Park Service-run, perfectly.
And just a few miles away, the most amazing views of Manhattan across the river, not to be missed at sunset as the lights go on! Including a moving permanent tribute to NJ Firefighters who reponded on 9/11. A happy accidental find!

I wouldn't describe it as a pilgrimage, but I have made trips to go see some sites that might fit within your description. Mostly when I was younger, I went to Yosemite National Park in California to see the Devil's Post Pile, to Sequoia National Park to see the giant Sequoias, and to the White Mountain to see some of the oldest White Pines on earth. I may have revered these sites, but I did not worship them.

I'd love to go to well known fossil beds and spend many weeks digging here and there.

Exploring, curiosity, study... I do it as much as possible

Great question, and great theme for a vacation, but I would have to think about it. All your mentions would be on the list of possibilities.

Worship of what does exist rather than what doesn't, if characterized as religion, makes them more wrong not less since they have just admitted that a true religion exists, but it's not their's (and they should really stay out of theological discussions) I've spent many days in awe at the dinosaur displays of New York and Denver's natural history museums.

Rather enjoyed the Eden project and would like to visit Findhorn. As a child my favourite place was the London Science museum and I’ve enjoyed other science museums too. But also like learning human myths and legends, which suppose could be considered anthropology.

That is a very interesting question.

Past visits:
The Royal Observatory at Greenwich
Kew Gardens
The stromatolites at Hamelin Pool
The Corinth Canal
Coral at the Great Barrier Reef
Clifton Suspension Bridge
Cornish engines
Snowy River power station

Future visits:
The Grand Canyon
The bottom of the Marianas Trench
The Giant's Causeway
CERN
LIGO
Jodrell Bank

On my bucket list is seeing places like Machu Pichu, Göbeklitepe, and the Nazca Lines. I won't because I don't have the money and I'm not healthy enough to visit them.

I am also hoping to be alive to go and see the rocket lift off for the first human Mars mission. I also want to be alive to actually see aliens publicly acknowledged and visit Earth.

I remember in the 70's when they first discovered the Terracotta Warriors, I went nuts. I wanted to see them when they toured the US, but was unable to.

Definitely natural phenomena. Visiting national parks like Yosemite, Sequoia and the Grand Canyon tops any man made structure.