What should I definitely do at Kearney, Nebraska when visiting during vacations?
I would recommend going to Pioneer Village. It's not in Kearney but in Minden, 13 miles away from Kearney. I haven't been there in decades, so better check online first. But it is a large interesting place .
In Kearney, Harmon Park, which adjoins the city swimming pool, is very nice... or at least it used to be. Cottonmill Lake is at the west edge of town or just outside of town, along Highway 30. Nice scenery there. People used to ice skate in the winter on Kearney Lake, on the northwest side, near Hillcrest homes. They have nice college campus. There's not really much to do in Kearney that I know about.
There is the "Arch" that commemorates the Oregon Trail that went by Kearney. It is fairly popular. East of Kearney is Fort Kearney, which was an important fort during the Oregon Trail period through the Civil War. There is Pioneer Village in Minden that is just a few miles east of Kearney. In Grand Island is the Stuhr Museum of the Prairie Pioneer that is popular. During the warmer months they have outdoor exhibits that are well regarded. There is a museum of art at the University of Nebraska-Kearney, and a very fine museum in nearby Hastings. Fonner Park in Grand Island hosts the Nebraska State Fair in late August. Grand Island, Hastings, and Kearney (the Tri-Cities) have great things going on with shopping and other activities.
Go to the Arch.