Today I started to play with the ultrasonic range finder. In the end, the following tutorial provided everything I need:
The sensor readings do not seem very reliable at the moment, but I think there is some loose contact on my breadboard (I did not yet solder anything yet). If I hold some contancts firmly, the readings are okay. The maximum distance I have tried was 4 meters, which seems to be correct. The nearest was below 1 cm, but the values varied greatly.
All in all I am quite satisfied, because my main aim with the sensor is to get some idea how far above ground the quadrocopter is. These values do not have to be exact, I plan to combine them with information from other sources (for example GPS, air pressure) in a Kalman Filter (or something similar).