These are worth the visit:
NASM- Washington- still the standard for the others.
New England Air
Museum- Windsor Locks, CT
It's a smaller museum, but they have a lot of interesting stuff, and it's more up-close and hands-on than others.
Kansas
Cosmosphere- Hutchison, KS
The focus is obviously on space here. They have a lot of genuine artifacts, including an Apollo capsule and things from the X-plane projects.
EAA
Museum- Oshkosh, WI
Surprisingly large and polished museum with a lot of things to see. Has a lot of hands-on stuff for the kids (and grownups!). I like the Pioneer Airport, a short grass strip outside the museum that is set up like an airport from the barnstorming days of the '20s, complete with several dozen aircraft from the era.1
Imperial War
Museum, London and Duxford
The Duxford annex is exclusively aviation, but the main museum in downtown London is world-class, and worth a visit. It's one of the only places I know of where you can see real V-1 and V-2 rockets.
Planes of Fame Air
Museum- Chino, CA
Lots of rare stuff here, even if the museum has a cluttered layout with amateurish displays. One thing I like about the museum is all the pieces that are just shoved into a corner unrestored. It's amazing some of the things you can see: '50s target drones, old Whittle-type engines just lying around, and the forward fuselage of a B-50.
There are several warbird restoration shops on the field, too.
USAF
Museum- Dayton, OH
A close second to NASM. Lots of rare stuff, and they're getting experimental prototypes all the time for display. They have at least one example of every aircraft the Army Air Corp, Army Air Force and USAF has operated or evaluated.