Completist (or completionist in the context of video games), as suggested by Andrew Leach, is the term I would use for someone who desires to do every last part of something. But your question is about what to call the person who tries to visit all examples of something, an activity I would call collecting in its broadest form.
Specific communities of enthusiasts may have their own terminology, of course, like peak bagging and highpointing among mountaineers. Unexpectedly, there seems to be no specialized term for visiting all the Major League baseball parks in the U.S., even though that is extremely common (I have probably half a dozen friends who have done it).
In this usage, collecting the experience of a place is treated much in the same way as one might collect physical items like postage stamps or coins, or collect the experience of viewing something in various spotting, watching, chasing, or other observation hobbies as fans or buffs might do.
This usage is not especially new. I found a reference from Cassell's Little Folks magazine, Vol. 82 (1915).
I have, indeed, heard of a grown-up man who says that he is very fond of "collecting cathedrals," and of another one who "collects battlefields." This, of course, does not mean that they put such things in a museum of their own, but only that their particular hobbies are visiting beautiful cathedrals and historic battlefields, and learning all that they can about them.
Some more recent examples include the following:
Collecting states is a family tradition. [Kearl, Holly, "American odyssey: 50 states in 31 years," CNN, May 12, 2014]
They sound like the fictional settings for fantasy novels or pulp adventures: the Maluku Archipelago, the Kingdom of Mustang, Peter I Island, the Republic of Bashkortostan…. But these are all real destinations, common entries on the checklists and wish lists of a new breed of traveler spawned by the Jet Age: the country collector. [Kennings, Ken, "The World's Most Traveled Man," Slate, Sept. 22, 2011]
Surprisingly few travelers get to 100 different destinations, Shaver says, adding: “Unless you really work at it opportunistically and knowingly, with a goal of getting a hundred, you’d miss or bypass chances to collect countries. [Sicoli, Florence, "Milestones: Collecting countries and experiences," FYI (Forever Young Information) Hamilton-Halton Edition, February 2011, p. 34]
I am something of a county collector myself, although unfortunately I have not been able to improve my collection for several years:
