What should I read to learn about the limits of free speech in the media?

Is it illegal to make allegations in the media when you can’t prove them? If it’s illegal, how does the law deal with indirect language like allusions.
What kind of speech is punishable by the law?

Illegal or actionable in civil court?

I suggest read this Wikipedia article on defamation, as it appears you’re asking about libel.

Note that free speech is a different subject.

According to the article from Wikipedia which you suggested, freedom of speech isn’t a different subjects.

And that’s exactly what I’m asking about. I want to read more on these limitations. I want to read more because the media is full of rumors and defamation and no one seems to care about these limitations.

Constitutional law: principles and policies by Erwin Chimerinsky

Look up public figure.

In the U.S., which is what I assume you’re talking about, there are virtually no limitations imposed on speech concerning politicians. The only bar - other than death threats or the the equivalent, which is different from what you mention - is actual malice. Showing that a media outlet broadcast something about a politician that rises to actual malice is nearly impossible. It would be political suicide even to bring it to court. All that would do is make the allegation the talk of every media outlet in the country 24 hours a day forever.

This has been true since 1964, so it’s old news. Almost no politician ever tries to sue and the very few cases that have been won are beyond mere rumor, such as this local one from 2000.

To the best of my knowledge, the only material it is illegal – as in a violation of criminal law – is that which is constructively treasonous, i.e., actively aiding an enemy of the U.S. Breaking a story about what troops the Pentagon plans to land where (as actually happened during WWII) would be an example, as would the (classified) specific techniques used to create a WMD. (The sort of general information that can be deduced from a good physics text would not fall into this category; the specific engineering requirements would. Gag rules on particular controversial court cases might also incur contempt charges. The courts take the frreedom of the pressw seriously.

What is being discussed above is different – what is actionable in a civil lawsuit. In broad terms, with rare exceptions “the gummint” cannot tell you not to publish something. But publishing a known falsehood with malice aforethought can lead to a lawsuit from the person being libelled or slandered.

Although defamation laws (including libel and slander) most directly answer your question, I just want to point out that there are a bunch of other free speech restrictions in the US. This wiki article is a more targeted summary:
United States free speech exceptions

Somebody correct me if I’m wrong, but I don’t believe the media has any special protections or restrictions in regards to free speech over any regular citizen, though they would certainly have better lawyers.