Mains doorbell to Arduino
I would use an opto-isolator between your doorbell and your Arduino. Make sure you get one that is suitable for the voltage and current your doorbell operates with. Measure those values with a multimeter if unsure.
If your doorbell operates on AC mains then you might want to use a rectifier circuit as well because if you just throw away the negative half of the cycle your Arduino might not "see" that the button has been pressed.

Detecting that someone pushed the button should otherwise be fairly straight forward. Just have a look at the Arduino "button" example.