The San Diego Natural History Museum has two of my favorite "exhibits" ever. They are what they call "living exhibits" and by that, they mean you can watch people performing tasks that you don't see often anymore. One is a man weaving -- he does everything by hand, and you can see what the whole arduous process entails and almost imagine yourself back in the day when everything was made by hand this way. The second is a woman who makes tortillas. You can actually buy her handmade tortillas (which taste so different from the ones you buy at the store, even the "handmade" ones there) as a quesadilla.
They do seem to have interesting "normal" exhibits here too -- not your typical boring natural history ones. Currently it is one on backyard insects, prior to that was one on chocolate (and its history). They also have programs sponsored by the museum, and they all sound so interesting that I can't decide which ones I'd like to sign up for -- they range from vineyard tours in nearby Mexico, to kayaking, to regional cooking classes, to mushroom hunting. I salute their program director who has certain kept things interesting and make learning fun.
On this note, I like how they have exhibits that are on "trial" and are looking for feedback on before they design the permanent exhibit. I think this is a great tactic to get the public involved.
The admission is a bit pricey for a museum I find -- I guess it is to encourage memberships.