Easy Door bell notification?

What I want to achieve...

  1. When someone presses my door bell(s) I want an SMS sent to my phone.

What hardware I have now...

  1. Arduino UNO (not internet connected)

Facts about my setup...

  1. My door bell is unplugged right now (no weird voltage circuit worries, basically just button press from momentary on button). I did this so that no one would inadvertently wake up my kids while they are napping. I intend to leave it this way.
  2. I have three door bell buttons and I would like to get a specific notification depending on which one is pressed.
  3. My house has internet both wired and wifi. Probably already assumed but just incase someone was wondering.

Things I would like to add after the doorbell project is complete...

  1. Sump pump water level sensor. This could be as easy as a closed circuit if water reaches the top of the crock. Upon circuit closure a SMS is sent specifically saying water has reached the warning level.

What do I need to know...
1. What add on board should I get to get internet connectivity to my uno? Or should I just buy another board with built in wifi/wired connection?
2. Do I absolutely need an IOT account with some company to send the SMS or can I just send an email to my phone number @ my cell provider?
3. Is there anything I am not thinking of?

Is there a low voltage power source available to each doorbell, or will they need to run on batteries? Is there at least a weak WiFi signal at each doorbell?

PaulRB:
Is there a low voltage power source available to each doorbell, or will they need to run on batteries? Is there at least a weak WiFi signal at each doorbell?

There is just a button with two wires at each door. Those run back to the main chime box. My idea is to take those to wires on each button and just connect them to the Arduino as open circuits and when the button is pressed it will close the circuit between two "analog in" pin inputs. Does that make sense?

SKINNYKID:
when the button is pressed it will close the circuit between two "analog in" pin inputs. Does that make sense?

That part of what you said doesn't, no. But at least we know you have wires going back to a central point. I will re-state my question then: is there, or could there easily be a power source at the chime box? Is there WiFi signal at that point? Or ethernet?

Assuming there is only mains AC and WiFi, I would abandon the Uno and get a Wemos mini. Also get yourself (if you don't already have) a largish breaboard, some small pushbuttons, solid-core hookup wire and a range of resistors so you can prototype the circuit and develop the sketch before deployment. When installed, you can power it with a basic phone charger through its micro-usb socket.

You can use your Uno to begin the prototyping until the wemos arrives. But you will find the Wemos much more convenient when building on the breadboard because, unlike the Uno, the Wemos mini can plug into the breadboard!

Oh, another question: do the doorbell buttons contain lights? Are they led or filament bulbs? Are they permanently lit?