I would like to know what the circuit of a door bell (front lighted button) no back door bell, how it is wired with four wires. Two solid blue and two blue/white. Four each on the transformer ane four on the bell transdusher. Two on the button, front door. How does this work?

Sounds like someone doubled up spare wires. Are there two wires per terminal?

The transformer has two terminals, one goes directly to the chime, the other goes to one terminal at the pushbutton, and the other terminal on the pushbutton goes back to the chime.

Should be as simple as that.