Man you should definitely have a look at HubDuino. I use them for my garage doors and the Arduino chips are so cheap e.g. around $5 AUD with a relay from AliExpress. I recently also purchased a temperature and humidity sensor which I plan to install to my garage door opener at some stage too. Anyways there is a button sketch that you could use as your doorbell then just use whatever doorbell/button that you like.
