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Does the U.S. Constitution's First Ammendment protect false speech? In other words, does it give a citizen or the press a right to spread falsehoods publicly?

Sometimes

In general, intentionally false speech gets less protection than other speech, and in some cases it is unprotected. The classic example of speech that is unprotected is "Falsely shouting FIRE in a crowded theater". Note that this is both intentionally false and highly likely to be seriously harmful to multiple uninvolved people.

On the other hand, the classic case of New York Times vs Sullivan said that, at least when the subjects were public officials (later broadened to public figures) it was not enough to prove simple falsehood in a defamation case, one must prove "actual malice" (an unfortunate term) which in this context means statements that are either knowingly false or are made with reckless disregard for the truth. The court in that case said, in effect, that if a newspaper had to be sure that its every statement could be proved true in every detail, it would be unwilling to vigorously report on matters of significant public concern (this is a paraphrase, I'll add a quote later).

Opinions are considered legally not to be either false or true. "President Jone is the worst leader the US has ever had" Is a statement of opinion, and so is not defamation.

Moreover, in political contexts, attempts to punish false statements of fact that are not defamatory have been held unconstitutional. One example was the "Stolen Valor" act, which punished falsely claiming to have been awarded a medal by the US armed forces. This was held to be against the First Amendment.

In general, regulation of speech (which here includes writing and other forms of communication) must be fairly narrowly drawn and must have good reasons behind them to survive a court challenge. How much so depends on the nature of the law, and particularly whether it is "content-neutral" or not.

Details and cites to come when i have a little more time.

Yes, under the First Amendment you may freely promulgate falsehoods in public. The only restrictions related to truth pertain to sworn statements (perjury), representations made in legal matters to the government, fraudulent statements (statements that you know to be false and are made to entice a person to engage in a contract of some sort), false (as well as deceptive) statements made in advertising, and defamatory statements. All political falsehoods are legally protected.