There are many usages for the very common verb get.
Let's get started!
You'd better get going.
We have to get moving now.
the sense is:
.15. To begin or start. Used with the present participle [or past participle]: I have to get working on this or I'll miss my deadline. [AHD]
It is a catenative usage, and more or less colloquial. Notice that there is an element of redundancy in 'Let's get started'; it probably sounds less formal / clipped than 'Let's start'.
There is another, similar-looking usage, the 'get-passive':
Do you think you might get shot?
They got married yesterday.
Here, the meaning of 'get' is 'become', or 'be' in the transformative rather than durative sense.
In your examples, 'This action got started' might be used especially in the US, but sounds unusual to British ears. It would be the passive, meaning 'was started'.
'We got started' sounds more acceptable in the UK, but now has the non-passive sense.
'The robot got started' is ambiguous.
'The action started / began' is the usual way this would be written, especially in the UK.