You might want to start by looking at the
GPS logger example sketches. Both the Geiger counter and the GPS modules send periodic data over a serial connection. Logging Geiger counter data will therefore be a similar operation, and a similar-looking sketch.
The Geiger counter serial data lines are at 3V, so you will need a level-shifter to get them to the 5V levels the Arduino uses. You could use this
level shifter chip, or this
level shifter breakout board. You should be able to install either on the logger shield prototyping area.
Then, it's just a matter of connecting the Geiger counter serial data lines through the level shifter to a pair of Arduino digital pins (e.g., 2 and 3, as in the GPS sketch).