I highly recommend getting everything working at your workstation before installing. You can simulate a button press by grounding the input pin and test the circuit with a multimeter. The meter should read 12V very briefly off the transistor and you should receive a text message!
If you're feeling creative, you could easily modify the code to ring the bell using a different rhythm.
Warning: The following steps involve potentially dangerous electrical work- if you don't know what you're doing, just stop.
First, shut off the power to your doorbell. Test the doorbell before and after and take all other necessary precautions to ensure the power is off.
Next, locate your existing doorbell transformer (usually in your basement) and disconnect the wires from it. If you're lucky, there will be two sets of wires - one for the push button and one for the solenoid. In a normal installation, the button closes the circuit and activates the solenoid to ring the bell. We are going to change it so the button is an input for the Arduino, which will then programmatically control the solenoid using the transistor circuit and external power supply.
Lastly, mount the board somewhere safe, connect the wires as show in the schematic and plug in the power supplies.
With the power back on, you should get a satisfying ding-dong, follow shortly by a text message.