Washington: go to the Air and Space Museum. They have an actual piece of Moon rock that you can touch! When I was there about 1992 they had a mock up of the skylab, lots of rockets etc. They also have the command module from one of the Apollo missions. I spent a whole day there.
Of course the Cape. You can take a bus tour around the launch pad etc. I made a big mistake though. The bus is double decker, the top has a glass roof. So I got in the bottom, I wanted to be the first out when we got to the pad so I could have a look around. Unfortunately, you just drive around the pad so I just got to see the cars driving around while the others could look up at the shuttle. The Atlantis was actually on the launch pad (they took it in the next day with a problem with a pump, hasn't changed much has it). The tour shows you a lot of things at the cape. Also they have a Saturn 5 lying on its side...this is the most ginormous, humungous thing I have ever seen in my life!
Of course, if you are going to Hawaii go to the big island and take a trip up to the scopes at the summit.
Not quite astro, but if you find yourself in Vegas, take the Star Trek ride at the Hilton (even my wife loved this one).
That's about all the "astro" stuff I have managed to do so far. I was hoping to go on the Sky & Telescope astro cruise from LA to Hawaii later this year, but time and money just wont fit it in
