I think that, rather than starting with the genes, you might consider starting with the alleles, or better yet, an explanation of the only two colors (pigments/melanin) found in chickens - black (eumelanin) and red (pheomelanin).
With the exception of the 3 pattern genes, basically all genes either intensify, extend, dilute, remove, or have no effect, on red, black, or both. How these genes all appear and interact with one another depends generally on which allele they they are found.