One of my favorite accounts is in Vodka Politics: Alcohol, Autocracy, and the Secret History of the Russian State by Mark Schrad. On Page 173, he talks about the raucous 1870s visit of Alexei Aleksandrovich (a Russian grand duke, uncle of tsar Nicholas II and commander of the entire Russian navy). Here's an excerpt:

"His American host — former Civil War general Philip Sheridan, famous for carrying out a scorched-earth policy similar to that of William Tecumseh Sherman's march to the sea — graciously obliged, going so far as to create "Camp Alexis," a veritable Potemkin village on the Nebraska plain. The "authentic" frontier conditions at the sumptuous camp included a brass band, the finest foods and wines, and crate upon crate of champagne. The once-fearsome chief Spotted Tail and his Lakota tribesmen were paid to act as stereotypical Indian savages, and Alexei was taught how to hunt on the open prairie by none other than Gen. George Armstrong Custer and "Buffalo Bill" Cody. Their "genuine" Wild West hunts were interrupted promptly at lunchtime by caterers with wagonloads of sandwiches and champagne. ..."

There's more of a description of this debauchery, which goes on for most of a page.

There was a long-standing European interest in the North America wilderness. In the 1830s and 1840s, the "American West" that they would have been considering as both pristine and accessible would not be the "West" that we know of today. Accessible would have made it nearly impossible to go west of the Rockies or east of California, and even the Rockies would have been a dangerous and difficult trip. So your Europeans would have been looking at the Old Northwest, the shores of the great Lakes and the hinterland that would have been relatively easily accessible from those places at the time. This is outside of my period, so I don't have any examples of European explorers/tourists.

An early example of a European explorer/tourist to the Great Basin was Sir Richard Burton, fresh from his exploration of the African interior. In 1860, he traveled to Salt Lake City and then across the Great Basin to San Francisco to board a ship back to England. He wanted to go to Salt Lake City because of his interests in religion and sexual diversity: the new Mormon faith with its doctrine of polygamy proved just the lure to attract the great explorer. He wrote about his travel across a great amount of unsettled and fairly wild real estate in his book "City of the Saints." He serves as a later example of Europeans exploring the wilderness of North America. And that's the best I can do to answer your question.