Must see sights in and around Amsterdam.

Personally, I would put the Van Gogh Museum on the must see list. I was vastly disappointed that I didn't get to see it last summer. It was closed because of the parade for the Dutch Football team. That was a once in lifetime experience, but I sure wished I had gone to the museum the day before. Be sure and include time for just wandering the streets / canals of Amsterdam. The architecture is really interesting. I also enjoyed a walking tour I did of Haarlem. You can pick up a pamphlet at the Tourist Office. I had great fun at a market on Saturday. It was in the eastern part of the city. I can get more specific if you're interested. There were strange things to buy and, of course, could get herring. I had visited the Anne Frank House on a previous visit. If I had had more time, I would have revisited it. I too visited Keukenhof. I was there in April for my first visit so it was a good choice. I'm not so sure if I would have gone at another time when the tulips were not blooming. Pam

I too recommend the Van Gogh Museum, as well as the Rijksmuseum and Anne Frankhuis. As Pam said, also give yourself time to walk around the city and/or sit at a cafe. Go to Dam Square and the red-light district. Chinatown is also cool.

It's outside of the city, but I did enjoy visiting Madurodam. If you're not famliar, it's a miniature Dutch city, google it to see.
I'd also recommend the Dutch War Resistance Museum.

You asked about outside of Amsterdam. You have a pretty nice country to explore. I can list several: Utrecht is a beautiful, refined, laid back, close town with a great railway museum the low level canals in Utrecht with split level shops check out Zaanse Schans for nearby, easy to get to, windmills and touristy clompen Kinderdijk is many more, much bigger, further apart, windmills set in wild land near Rotterdam the town of Leiden has a medical history museum, gardens, windmill, and is the nearest town to Keukenhof Delft Den Haag There really is a lot to do and see beyond Amsterdam. I haven't mentioned the genuinely excellent open air museums of the Zuider Zee at Enkhuizen and National Open Air Museum at Arnhem which are a little further away but still easy to get to. Wish I were going in your place. Hey, how about renting a bike near Keukenhof and cycling through some of the fabulous bulb fields. How about a walk in the dunes beyond Haarlem? Or Haarlem itself? Happy planning...

Make time to visit the Hermitage Museum; now Amsterdam's largest art museum. IMO the most underrated sight in the city is the Tropical Museum with beautiful exhibits about 3rd world countries and cultures. Check out http://www.iamsterdam.com/en/whats-on for details on all the temporary exhibits.

Re Nigels suggestions: outside of town Zaanse Schans (my sp is horrible) KinderDijk are worth seeing for "typical Holland", Maarken and the fishing villages where they wear traditional dress (for the tourists?) Alkmaar cheese market is a fun experience; also you can train to Delft or even Brugge

I was in Amsterdam in spring of 09 and my favorite places to visit were 1. Keukenhof gardens only opened when the tulips are in bloom. Plan on spending a whole day visiting these beautiful gardens. 2. Van Gogh museum I am not normally a museum person however the VG museum is well laid out and the main floor is arranged chronologically and easy to follow with RS guidebook. Buy you tickets in advance to save time. 3. Anne Frank House tour Buy tickets in advance for this historic house. 4. Delft Go see the Royal Dutch Delftware Factory and watch workers hand paint the blue earthenware. 5. Aalsmeer Flower Auction 6. Rijksmuseum Tour Be sure and eat at one of the "Indonesian rice tables" which is a specialty in Amsterdam.
Z-S was not worth the day visit as very touristy.

We loved Haarlem, about half hour by train from Amsterdam. Lovely square with beautiful Cathedral. Loved the Ten Boom Museum, look it up on internet, interesting WWII history, true story. Enjoy Amsterdam. We took a canal cruise from Boom Chicago nightclub, very small boat, only about 6-8 people, brought our own cheese, wine, was great.