I think you might get a different answer from every American you ask, but I am American, and on the weekends we like to go on hiking trips with our dogs or short road trips to nearby towns/cities, where we usually see the sites, shop, and try new restaurants.
On more mundane weekends, we run errands (shop for necessities), see movies, dine at familiar restaurants, or sometimes hang around at home watching movies; we might be alone (my husband and I) or with friends. We very rarely visit family on weekends since we do not live near our families.