Visiting stately homes is a fairly common family day out in Britain, especially since so many are open for public viewing thanks to the National Trust and similar organisations. I had always assumed that this started once organisations like the Trust took over these places, until I recently read Pride and Prejudice.
In the book, which predates the National Trust by about a century, Elizabeth and her aunt and uncle are on holiday in Derbyshire and visit the gardens and house at Pemberley. They are even shown round the house by the housekeeper and shown pictures of the family who live there. Would this be normal behaviour? Were these houses considered somewhat public or would it be strange to just knock on the door and ask to have a look around?
There are travel writers in the 18th century that discuss visiting stately homes (in addition to cities and other more far-flung locations) as part of their travels.
Travel for pleasure in general was coming into vogue. At the time, it was generally expected that as long as you looked sufficiently middle class, you would usually be invited in and given a tour. According to the British Library, to do otherwise was to be looked down upon by other travel writers of the day. Some houses, like Chatsworth, were almost as popular if not more so than they are today, and were forced to create more rules or specific open days so as not to be inundated with guests.
Some owners enjoyed it, even turning a profit from it by selling guides to their art collections or selling food/mementos. However, it could be a burden. In the US, George Washington’s Mount Vernon became a must-visit to a near-constant stream of guests and pilgrimage-takers, all of whom had to be formally greeted, given a tour, dined with, and (in many cases) given a bed for the night.
Most travel writers are focused on the homes rather than the occupants therein. One well-known traveler, Celia Fiennes, presented it as her duty to learn more about her country, and this seems to have been a relatively common view. And indeed since tours were sometimes given by servants and the owner might not be in residence that time of year, one could theoretically visit a home without encountering the family.
Sources:
The Illustrated Journeys of Celia Fiennes 1685–c. 1712
British Library, Domestic Tourism
Mount Vernon (Specific stats from Visitors to Mount Vernon)