You do mean goldfinches, not Gouldian finches? I didn't know anyone bred goldfinches. I've never heard of this, but would be very interested to hear about it if you have experience. Which goldfinch are you wanting, American or European?
I think that goldfinches are one of the protected songbirds in the US and if I am correct, it would be illegal to keep and breed them without a special license.
Feathers (our member here) will know for sure about the licensing.
Annie