: How do I write a story within a story? How can a protagonist, being a writer himself, share a full short story he has written with the readers within a short story? What are the techniques

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First, a definition
What you are asking for is called a frame story. Imagine your novel as a picture on a wall. The protagonist of your novel and his activities set the frame of the picture, the things that are happening outside. The short story he has written is the picture itself, the thing that happens inside the novel.
Second, do research
Now that you what you're looking for, you can do some basic Google searches for "How to write a frame story" and get a more general answer to your question. Here are some good resources I found while browsing:

LiteraryDevices lists many good examples you can read for research.
Sophia.org has a tutorial with visual depictions.
WriteOnSisters suggests some tips you'll need to keep the story straight.
And on our very own Writers.SE, Paul A. Clayton wrote this great answer to a related question.

These are by no means the only good resources, simply the first three I found in a basic Internet search. Once you get a general overview of your problem, find some in-depth writing books that discuss it. Look at the great wealth of books on writing that exist. Pick some and read them, taking notes the whole time. This should be done concurrently with my next step...
Third, read like a writer
A key step is finding literary works that display the technique you'd like to use and then reading them while taking notes on how they effectively accomplish it, exactly as Mbakeranalecta suggested. You can get many suggestions here and from the above links.
It's not enough to just read the book. You have to read like a writer. Read it once, to enjoy it and to get a general feel for what happens. Then read it again. Take notes:

Where is the frame story written? Where is the interior story written?
How much space does the author develop for each part?
How often do they switch between the two stories? At what points do they switch - at a climax? After resolutions?
What literary techniques do they use to mark the difference between the two stories? Do they use different perspectives, as you suggested using? Do they use different tenses (past vs present)?
What do you love about their work? What do you wish they did differently?

Ask yourself all these questions and more. Ask yourself what you need to know to reproduce this in your own work, and then find that information in the established literature. Read like a writer.
Fourth, write
This may seem an obvious step, but many new writers struggle with it. In order to be a writer, you have to actually write. It is essential to finish what you begin. Finishing your works will teach you far more than I ever can convey on this Internet forum.
I will say that I like your idea to have the frame story in one perspective (in this case, first person limited) and the inner story in a different perspective (in this case, third person omniscient). I think this will do a great job of seperating the two stories from each other and, coupled with other literary techniques, should make it fairly easy for audiences to follow the transition. Other, more specific tips, you will have to discover for yourself.
Good luck!

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Is your idea like Russian nested "Matrushka" dolls, a story within a story within a story? If so, it seems to me that the recent movie "In the Heart of the Sea", a story about James Fennimore Cooper, the Nantucket whaler "Essex", its captain and crew, the great white whale, and the Essex's survivors, is put together like those Matrushka dolls, and seems to me to provide a good example of what you seem to me to be proposing. Consider you, the screenwriter of the film as the outer protagonist relating the story of his protagonist James Fennimore Cooper relating the story of how he came to write 'Moby Dick', his story being from an outer protagonist relating the core story related by another, a third and final, inner protagonist. It is the third protagonist's story that contains the core/complete story that the screenwriter (you, if you will) has put together and has slowly revealed in the movie. However, you can also put yourself in the place of the screenwriter's James Fennimore Cooper, and relate (in Cooper's words, so to speak) the story told to him by the sole surviving member of the Essex's ill-fated crew.

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The answer is somewhere around, like what mbakeranalecta and White S. wrote:

A small story can be the same like everyday life short stories. Your protagonist may pick up a diary or browse a blog and read the log of a day or part of it, or getting involved in a letter, report. He can do research to look for specific sources or to confirm hypothesis, and so the details will become important.
However I do not know much about your story, I would suggest to avoid turning to "reading a book-like thing".

The separation will be done by you, setting the atmosphere, maybe keeping a suspense with time-to-time throwing in the thoughts or feelings of the protagonist, maybe getting a conclusion at the end (as a debrief for the reader).

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Well, that would depend how long this short story is. But to me, the first idea that comes to mind is to show this short story by having a character actually reading it, perhaps within the framework of your author having asked for some feedback from a friend or even his publisher. Either that or make it your first chapter showing the issues he has with it (assuming there are any) perhaps by using an opening paragraph like describing what the book means to him and asking the reader what they think of it before showing it.

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