Welcome!
First, I would divide this into two separate projects, the cameras and the doorbell.
You can easily tie everything together so that the doorbell being triggered starts the cameras or the cameras being triggered causes a chime or spoken notification to sound the various rooms in your house. So I wouldn’t worry about that part.
1. Cameras
People use several different brands depending on their own goals. Arlo is very popular and has an official integration with smartthings, so most people start there.
( in mid 2019 smartThings introduce their own inexpensive Wi-Fi camera, but it’s indoors only. )
https://www.smartthings.com/products/-/filter/categories/cameras-and-doorbells
Ring cameras and the doorbell also have an official integration, but it sounded like you wanted something cheaper than those.
Blink used to be very popular, but when they were bought by Amazon the smartthings integration was broken and never repaired, so now the only integration which is available is through IFTTT.
Anyway, just search the forum and you should find a lot on cameras. Or hopefully others will chime in here.
2A. Aeotec Doorbell 6
Aeotec has just released the newest version of their doorbell in both the US and the UK. I haven’t seen any reviews on it, but the previous version was popular and I expect this one will be good.
The chime unit is very similar to the aeotec siren, but with a wider choice of sounds. And it comes with its own matching doorbell button.
The chime unit is Z Wave, the button is on a different frequency and is paired directly to the chime unit. So it will work as a basic doorbell even if your SmartThings system is not working.
The previous generation was set to typical doorbell loudness, 85 dB, but they got a lot of requests for a louder one so now it’s the same volume level as the siren, 110 dB.
You can buy two additional buttons and sync them to the same chime unit and use different sounds for each if you want.
https://www.amazon.com/Aeotec-Doorbell-Outdoor-Wall-Mounted-Wireless/dp/B07PB2HN6D/
you would need the following custom code:
Anyway, that would be a simple doorbell solution. In this case you are replacing both your existing chime box and your existing doorbell button.
2B. Nexia doorbell sensor or similar do it yourself project
The other approach is to keep your existing doorbell button and your existing chime box and use a sensor to detect when that chime box sounds.
Nexia has a Z wave device made for exactly this purpose. You wire it into your existing chime box. It sends a message to the hub when the chime box sounds, and then you can use that message to trigger whatever other actions you want.
https://www.amazon.com/Nexia-Z-Wave-Doorbell-Sensor-DB100Z/dp/B017VVO9QQ/
It’s quite popular. There are also a whole bunch of different do it yourself projects which use different kinds of sensors to capture the same event. For example, with some doorbells you can just put a vibration sensor on the chime box. Or you can get more complicated. To see what other people have done in this regard, go to the quick browse lists in the community – create it wiki, look down near the bottom of the page for the “project reports“ section and choose the “doorbell“ list.
3. Getting notifications throughout the house
There are many different ways to get notifications throughout the house when a doorbell event occurs.
You can use a chime device like zooz or Dome chimes.
You could have a light change color.
You can have an echo dot speak custom phrase like “someone is at the front door.“
There is a how to article in the community – create a wiki on generating a sound when the door is opened which lists most of the options. In your case, you would trigger off your doorbell event, not off an open/close sensor, but it’s the same idea. 