Seattle is home to the largest aerospace manufacturer in the world, so it stands to reason it is also home to two separate aviation museums. See the main Seattle page here
Both are run by private companies but showcase a lot of Boeing's products and history. For a more Boeing-centric experience, visit the Future of Flight, which includes a tour through the Boeing factory. For a larger exhibit with international flavor and a section dedicated to space, go to the Museum of Flight.
Future of Flight Aviation Center & Boeing Tour
Have you ever wondered how the Boeing jumbo Jets are built? If so, this museum is a great place to glimpse how models 747, 767, 777, and 787 (the Dreamliner) are built.
This museum is located in the city of Mukilteo (next to Everett), about 25 miles north of Seattle. Before the tour, you get to visit the main Gallery (Learning Center), which has some informative exhibits and hands-on experiences.
You can get into the cockpit of a 727 plane, take a simulated flight, inspect GE and Rolls-Royce engines, and many other exhibits. While visiting, you can also design and print your own airplane image or visit the "How Planes are Made" family zone. 
Interesting Fact: The Boeing assembly line at Everett is the largest building in the world by volume, with 472 million cubic feet in volume.
After you have completed your indoor activities and exhibits, take the elevator to the observation deck, where you can view an active runway and Boeing factory buildings.
The tour is about 90 minutes and takes you through the Everett building and the largest assembly building in the world. You get to see airplanes moving down the long assembly line in various stages of completion.
Tip: Photography and electronic devices are not permitted during the tour. Make sure to place your belongings in the free lockers before you start - and remember your code.
It is a brisk walk, so it is not recommended for people with mobility issues. But don't let that stop you from enjoying this tour. The museum offers two tours a day that meet ADA requirements. You just need to call two weeks ahead of time to reserve it.
Interesting Fact: According to the Guinness World Records, the mural on the six factory doors is considered the largest in the world.
During the tour, you also get to visit the airfield and watch planes taking off and landing. The tour is very informative and engaging, even for children. The tour guides are Boeing employees and know their stuff. It was a nice touch when the guide pointed to his workplace in the plant.
Interesting Fact: The Boeing Company, founded in the Seattle area by William Boeing, started as a boat company.
The museum also has two gift shops and a café. One store is at the front of the tour and it carries Boeing items. Another store sells more traditional science gifts. For a free souvenir, go back to the gallery level. Take a free photo at the booth and return later to select your background and email it to yourself. You can also purchase a hard-copy photo at the Boeing store.
Read more about Future of Flight on their website here.
Museum of Flight
Museum of Flight is another must-see museum for space and aviation buffs. This one is in the city of Tukwila, south of Seattle, and comprises several buildings and a large outdoor exhibit.
One building is dedicated to space exploration. Inside, you can climb into a Space Shuttle model, browse through Apollo engines and model rockets, or visit the Space Flight Academy. When you finish the space exhibits, take a pedestrian bridge to the main building with a large aviation gallery. The gallery showcases airplanes through the ages, from the historic flight of the Wright Brothers to recent advances in space flight and all the innovations in between. Take the elevators to a balcony overlooking the entire exhibit for a better view of the whole gallery.
Another building tells the story of Boeing from 1916 to 1958. The first floor is dedicated to William Boeing and his pioneering decades in building this company that became the dominant aerospace manufacturer.
Interesting Fact: In 1915, William Boeing dropped cardboard "bombs" on a crowded California-Washington football game to wake up Americans to the danger of foreign attack.
The second floor explores the company’s shift from small craft to multi-engine airplanes, concluding with building the 707 jetliners.
Outdoors, you will find a large covered area with many aircraft from around the world. This exhibit has the only Concorde plane in the US. You can climb in and experience the seating plan and the complex cockpit. Also outdoors, you can find a Boeing 747, the first jet Air Force One, and many military jet fighters and bombers. The museum offers several tours and experiences.
You can go on a one-hour Boeing Field tour. The tour takes you both inside the historic airport and along the runway. You can also go on docent tours of the museum for a more in-depth experience of learning and entertainment. Finally, you can enjoy a 3D movie or take a simulator flight for an additional fee. During your visit, you can spend some time at one of two cafés to relax and enjoy some snacks.