Aside from the formal/informal distinction, there is a slightly different meaning between start and get started. Get started phrase begin a task, endeavour, or process For example, let's start the engine and see if the car won't make that noise again. as for your two sentences, i agree that let's get started on building this table sounds a bit awkward, but i might say let's get started on this table just as easily as let's start building this table. more on that in my answer below.
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You (had) better get started if you want to finish on time
To begin an important period in one's life or career newlyweds who are just getting started on their lives together
The form get started has the bare form of the verb get, used for present tense and as a bare infinitive (without to) The verb let's is a contraction of let. Getting started implies a description of the process, get started is a suggestion to the reader to do so, obviously to be followed by instructions. 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' In let's get started, the starting point is in view and let's get going, you are on the starting point already Moreover, there is a sense of extra involvement abundantly made clear by the sentence, let's start going.
From that point things started to getting complicated
Which of these sentences would be correct, if i want to imply that something happened and things are not in order anymore? Is there any difference between these two sentences We're about to get started