|Ancient architecture abounds. The old town is incredible. There is an old Jewish cemetary (you can only visit during daylight hours) with the tombsones stacked very close together. The Charles bridge is loaded with beautiful statuary. (Also, the first Mission Impossible movie had a scene shot beneath this bridge). The main square in the old town section, with a statue of Jan Huss (a missionary that attempted a form of protestant reform about 200 years before Martin Luther) is always loaded with people.
There is a wonderful clock tower with great ormanetation. Prague is the hometown of Kafka as well as many great artists.
Cathedrals abound - all of them centuries older than the United States.
Cross the bridge and leave the tourist areas to find some wonderful "local" shops and businesses.
If you're into the beer, try Kozel on tap and secondarily try the Pilsner. Kozel is not available in the US, and The Pilsner here is not nearly as good.
Most importantly - take any opportunity to get to know the people. Don't try real hard to get the language; they will tell you only Czechs can pronounce it correctly. But they are wonderful, freindly people, and if you can find interested students, they love to practice their English. The Economics University is located in Prague and the students there are the future leaders of this new/old state.
Above all, bring back pictures and stories. I probably will never get to go back, and I'd love to hear about your visit.