Anyone mentioned Vondel Park. Nice green open space down the southern end of the city, which on a sunny weekend has all sorts going on and lots of 'interesting' folks to see.
Leidseplein, touristy square, but nice to sit and have a beer or coffee and people watch, also Rembrandt Plein. Rijksmuseum, huge building down near Vondel Park, very good history of NL.
Someone also mentioned the Joordan area, check out Lindengracht for the market mentioned (l lived on that road for a while). Some nice bars and restaurants round there. Just wander around the Prinzengracht, lovely canal road and marvel at the architecture, bars, boats etc.