New England Aquarium - Full of fish, as all aquariums should be, and in a great location right on the Harbor and the Rose Kennedy Greenway. Being on the water enables them to not only do the traditional aquarium thing, but also to do whale watches and other kinds of marine-biology centered boat trips from right at their front door. The central giant tank has a great diversity of creatures, and the resident penguins hanging out are such characters and fun to watch. They have rotating exhibits about all kinds of marine life, and then upstairs is a shallow pool where you can touch (with supervision) some of our local sea life (horseshoe crabs, sea stars, etc). They also have an IMAX theatre, some of what they show is educational, some of it is totally not. It is expensive to get in, but the Boston Public library does let you borrow a pass for 4 from any of its branches.
Museum of Science - The permanent things include kinetic sculptures to teach physics, live animal exhibits about biology, a butterfly greenhouse built into the top floor, exhibits on the sun, geology, dinosaurs, a lightning show, and way more things about how the world works, too many to name right now. Traveling exhibits have been about Star Wars, time travel, the whole gamut. There's also a planetarium that does laser shows along with the regular night-sky stuff, and an IMAX theatre that's on a dome, much more immersive than a flat screen. Members here get special benefits too, a library, cheaper parking, and invitations to special events at the museum, especially the 4th of July fireworks viewing party on the roof of the parking garage.
Harvard Museum of Natural History/ Peabody Museum - Tucked behind the main yard on Oxford St, this is an old school natural history museum with lots of taxidermy animals and skeletons, which is either grotesque or morbidly cool, depending on how you look at it. The blue whale skeleton hanging in one of the main halls is amazing. There is a great fossil and mineral collection and also an incredibly beautiful larger than life glass flower collection. Attached to the natural history museum is the Peabody which has exhibits on Mayan culture, some Pacific Island stuff, and North American First Nations. Free on Sunday morning and Wednesday evening to Massachusetts residents.