Unscrew the doorbell button cover screws with a screwdriver and lift the cover off from the button so the wiring on the back of the button is exposed.
Set the wire tester to test for continuity. Hold one probe on each wire screw on the back of the doorbell button. Press the button and check the multimeter on the tester. The needle on the multimeter should move if the button is functioning. If the needle does not move when you press the button you will need to replace it.
Loosen the screws on the back of the doorbell button with a screwdriver. Lift the wires off of the screws. There is no danger from electrical shock, so you can handle the wires safely.
Make a loop on the ends of the two wires on the replacement doorbell button. Hook the wires around the screws in the back of the replacement button and tighten the screws. Replace the doorbell button in place on the house and drive the screws that hold the cover in place back in with a screwdriver.
Test the doorbell chime if the doorbell button was in working order. Remove the mounting screws in the doorbell chime with a screwdriver. Lift the chime from the wall.
Switch the voltage tester to the voltage setting. Hold the tester probes on the wiring screws on the back of the chime. The multimeter on the tester will indicate a flow of voltage through the chime. If the needle does not register any voltage flow, you will need to replace the chime. With a screwdriver, loosen the screws on the back of the chime that hold the wires. Lift the wires off the screws. Form a loop on the ends of the replacement chime wires and loop the wires around the screws. Tighten the screws.
Hold the chime in place against the wall. Drive the mounting screws back in place with a screwdriver. Push the doorbell button to test the doorbell. If the doorbell rings, the repair work is complete. If the doorbell button does not function, it is the transformer. A licensed electrician will have to repair or replace the transformer.