Starting up Mewgenics, and I have no real idea how breeding works and the best ways to handle it

Pretty much exactly what it says on the tin, really; I just defeated Spinnerette for the first time, and in spite of that I'm not too sure the best ways to breed my cats. Right now I just have my most recent victorious cats in the attic (the "winner's room", as I have started to call it), but nothing truly productive seems to be happening up there, and the resulting babies aren't that powerful. Am I doing something wrong, or do I just not have access to the best tools yet?