"This takes place in the Summer.  In the southern sky.  And slow and sure enough you rise and set and rollercoaster in the depths of night.
And night goes by in blue.  While you dolphin through."

The first part of this Kleeman and Mike episode refers to the constellation Scorpius, which is visible in the southern sky primarily in the Summer (Northern hemisphere).  As Scorpius "moves" across the sky throughout the course of the night, its angle and height in the sky changes giving it an appearance (at least to me) of the way a dolphin leaps out of the water.

Scorpius is one of the best constellations, so look for it.  In June it rises in the southeast around 8 pm, is highest in the sky around midnight, and sets in the southwest around 3 am.  In August, it rises around 4 pm, is highest at 8 pm, and sets around 11 pm.

600,000,000 miles.  This number assumes Kleeman and Mike are looking at a star in the Scorpius constellation called Antares.  Antares is a "red giant" star and as such is much larger than most stars.  Our sun by comparison is around 865,000 miles in diameter.

references:
Kerrod, Robin, The Star Guide, Macmillan, 1993

http://www.space.com/spacewatch/scorpion_020726.html

http://antwrp.gsfc.nasa.gov/apod/ap980726.html