You're looking for something to do Saturday that isn't the park again. The natural history museum is right there. You assumed it was expensive. It's free the first Sunday of every month. You had no idea.
The majority of museums, science, natural history, art, children's, aquariums, offer at least one free admission period per month or per year. Some do free evenings. Some are free all the time for kids under a certain age. Some offer free passes through the library.
The easiest way to find out: go to the museum's website and look for "free admission," "free days," or "admission." Most list it clearly. If not, call the box office, front-line staff know their discount programs and will tell you immediately.
Many public library systems partner with local museums and attractions to offer borrowable passes, free or discounted admission in exchange for using your library card. This is worth checking before you plan any trip to a paid attraction.
Search your library's catalog for "passes" or "museum passes," or call the reference desk and ask what passes they have available. In some cities you can check out passes online. Availability varies, and popular ones get booked, plan ahead for busy weekends.
Children's museums, hands-on, built for kids, usually a full afternoon's worth of activity
Science centers, interactive, works for all ages, often has planetarium shows
Natural history, dinosaurs and animals work at every age, surprisingly well
Art museums, better than you think for kids if you let them lead and don't force it
History museums, local and regional ones are often free and undervisited
Aquariums, expensive to buy tickets for, but often part of library pass programs
Zoos, similar; check for library passes or free days
If you're within driving distance of Washington D.C., all Smithsonian museums are always free. No catch. The National Air and Space Museum, the National Zoo, the Natural History Museum, the American History Museum, all free, all the time. A day trip to D.C. with kids costs nothing beyond gas, food, and parking. Plan a visit at si.edu/visit.
Free days are often busier than regular days. If that's a concern:
Go early, the crowd builds through the afternoon
Weekday free days (when offered) are almost always calmer
Check if timed-entry passes are required, some museums require advance reservations even on free days
Pack snacks and a water bottle. Museum food is expensive and you'll spend more in the café than you saved on admission if you're not careful.
Don't over-engineer it. Kids don't need a guided tour or a lesson plan. Let them move at their pace, stop at what interests them, and skip what doesn't. The goal isn't to cover everything, it's to spend two hours engaged in something together.
The exhibits will surprise you. So will your kids.