What are the best suggestions on what to see and do to make the most of Prague in three days (really one full day and two half days)?

http://www.prague.net/itineraries-in-prague
this should give you some idea of what you can fit in.I am just bck from Prague a few hours ago,but did not spend much time in the tourist areas.

A poster above suggested the Sandeman's "free" tour - and I totally agree. As first time visitors to Prague, this was a terrific 3 hour walking tour and a great orientation to the city and it's many architectural treasures. We also went on the Castle tour immediately after which made the most of 1 full day in Prague. You would be very hard pressed to see and actually learn as much on your own in a day. After that you have many interesting things to choose from, Art Nouveau lovers will love the tour of the Municipal House, among the many delights inside is the Alfons Mucha room. We both found the museum of Communism fascinating, a look into life behind the iron curtain. The Jewish quarter is beautiful, get one of the Jewish Museum tickets and visit at least the Pinkas Synagogue which exits through the Old Jewish Cemetary. Get there early as it gets very crowded. I did a trip report you might be interested in from a first-time tourist POV - click my name for the Trip Report link. You will love your time there!