The area around the '''Frauenkirche''' is called the '''Neumarkt''' and has some good shops and restaurants. On the southern end is a small alley called '''Muenzgasse''', at the lower end (towards the river) is a bratwurst stand that has some of the best grilled brats that you can find in Germany.
Up the stairs from here is the '''Bruehlsche Terrasse'''. Head towards the left (towards the old stone bridge), and you'll find the '''Royal Palace''', the '''Royal Church''' and further down the '''Semper Opera house'''.
Crossing the square in front of the opera house, you'll find the entrance to the '''Zwinger''', a baroque palace areal with different museum and exhibits in its wings. You can use the stairs on the northern end to get up on the upper level to find the fountains and sculptures hidden in the corners. The southern end has a Glockenspiel made of Meissen Porcelain, you'll hear it while you're there.
Walk through the gate and back into the old city, and head back towards the church. Along the way is the '''Fuerstenzug''', a 300 ft long mural entirely made of porcelain that shows all the ancestral portraits of the Wettin Dynasty, the ruling family of Saxony in these days. The creator of the mural has a cameo in the procession, it is the last figure depicted.