The example I use is New Orleans, but many similar activities, and even some not available in New Orleans, may be available in the city you are planning to visit. For example, New Orleans has no mountains, nor castles, nor covered bridges. Each city is unique, so look up the city of choice before you visit it. This is just a sample of what kinds of activities you might find. But, it has a long history with elements from different parts of Europe, Africa, South and Central America, and Asia melded into one.
New Orleans is a place filled with history. There are museums to be attended, including the Cabildo, a museum with artifacts dating back to French rule of the area, the D-Day museum, which has exhibits related not only to D-Day, but all of World War II, the New Orleans Museum of Art located in a park with its sculpture garden that is free to roam through, and the New Orleans Mint that once minted coins in gold and silver.
The shops along Magazine Street, and in the French Quarter, include many antique shops for those who like a glimpse of the past. The city was ruled by France, then Spain, then France again, then sold to the United States as Part of the Louisiana Purchase.
There are some places where one can even see the Mississippi River.
In addition to the museums, and the activities I will get to shortly, there are festivals. Many weekends have one or more festivals going on. The larger ones include Mardi Gras, a multi-day event, and three music festivals Jazz Fest, Voodoo Fest, and the French Quarter Fest. Other family friendly festivals can be found during the year.