Prague is wonderful, not sure how long you're there for, but here's a few recommendations...
I'd second walking out to Charles Bridge and up to the Cathedral, it's a brilliant city to walk around (and I've also felt very safe each time I've visited).
Hot chocolate and crepes at the Art Nouveau style Municipal House are definitely recommended.
The Maly Bhudda restaurant (up near the castle)
http://www.malybuddha.cz/
is a tea-house which serves fresh oriental food if you get fed up with beer and stodge. The Ginger tea is yummy.
For Czech food and beer served next to a roaring fire
http://www.pragueexperience.com/places.asp?PlaceID=366 this place is great just to realax and warm up in (though slightly pricier than some pubs). It's known to us as "Pub Lard" because they have lard on the menu. We thought it was a language mistake so we ordered it. Nope, it was lard, with bacon bits in
Good food at good prices and interesting location also to be had here:
http://www.prague-life.com/eat/restaurants_details/66-Klub_Architektu
Wemceslas Square looks wonderful at night but cafes and pubs around it are expensive and low quality (I wasn't approached by any hookers!).
If you aren't afraid of heights then climb up the Old Town Hall to get a smashing view of the city, particularly the pointy, gothic Tyn church across the way at Staromestske nam.
If you're interested in the history the Josefov area and Jewish cemetery are sobering, but interesting.
There are loads of interesting and beautifully decorated churches tucked away in squares. We tried to visit all the ones which contained dried bits of saints or severed (twig like) arms etc which had been lopped off thieves 300 years ago and put on display.
The trams and buses are reliable and cheap. If you're travelling from the airport, try to get the bus from the airport to Dejvicka underground station - it's the cheapest way. Otherwise there are fixed fee minibuses from the airport. Taxis have a very bad reputation for fleecing tourists.
Above all, enjoy it.