"get started" is a common verb in English. In fact, there are hundreds of constructions like this: get + past participle So you had better learn it!

"Let's start" ... a simple invitation to begin doing something together. "Let's get started!" ... the same meaning, however much more emphatic ... with a sense of impatience or urgency about it ... because once you are in the state of doing something you have 'already stared it' ... so the invitation is to 'hurry up and get ourselves into that state'. The latter would be used in a situation where the lecturer walks into the room a little late .. or a meeting you are attending has a delayed start, perhaps due to social prattle ... and the lecturer or chair of the meeting will say .... "Okay, let's get started!!" or expanded ... 'Okay, enough of the delay, let's get started'

In a language there are often many ways of expressing something, ways that are all correct. The minor differences lie in small shades of meaning. I would say 'let's get started' focusses the attention on the act of beginning, more so than in 'let's start', which is more neutral.

That's a very good question Grazia, and I can't think of a straightforward answer necessarily. However, remember that -to start is a regular verb and started is not necessarily only the past participle. The simple answer here is that the verb is a phrasal verb, -to get started. This falls under the long list of expressions with the verb -to get which is complicated all on its own. to get started is the same as -to start but just a common expression that we use in English. I hope this helps.

Thank you!