@Keo,
Take a look at my ST_Anything library. You could very simply treat your doorbell button as a contact sensor and then use standard SmartApp/Dashboard logic to send you a Push Notification any time the sensor is tripped. You can customize my sample to eliminate most of the example sensors in both the sketch and the Device Type code.
The bigger issue is whether or not you want the existing doorbell to work still. Doorbells use a 15VAC power source. You don’t want to fry your Arduino in the process. You may need a relay to switch the 15VAC to the actual doorbell. You will probably lose the light in your doorbell button though…
Here’s my library…