Cornell's website http://www.allaboutbirds.org/ writes that "paired-up goldfinches make virtually identical flight calls; goldfinches may be able to distinguish members of various pairs by these calls."
The female chooses her mate based on his bright coloring, song, and impressive flight displays. During courtship, the birds will fly in circles, with the male warbling throughout the flight.
After a male has been chosen, nesting territories are selected. Males will flit from one perch to another and warble around the perimeter of his territory. They will also perform flight displays. Look for the males as they fly in an exaggerated version of their normal undulating butterfly flight.