George Washington died on December 14, 1799. If I am a poor farmer in Georgia at that time, how long would it take for me to learn about Washington’s death and in what ways did this news move through the country?

George Washington died on the evening of December 14, 1799, shortly before midnight.

According to the article "The Death of Washington" by William Buckner McGroarty (The Virginia Magazine of History and Biography, 1946), the Alexandria Gazette was the first newspaper to break the news, being just seven miles away from Washington's home at Mount Vernon. They were informed sometime the next evening, and they ran a brief item the next morning, on December 16, 1799:

"It is our painful duty first to announce to our country and to the World the death of their illustrious benefactor, GEORGE WASHINGTON."

News almost certainly reached Washington, D.C., the same day. Newspapers in that city reported it the following day, and the day after that, then-U.S. House member John Marshall made a speech about Washington's death on the House floor. This was the day on which Washington's private funeral was held.

The following day, on December 19, Washington's personal secretary Tobias Lear made the announcement official in a letter sent to President John Adams, a letter dated December 16. Adams made several brief statements to honor Washington. On December 24, he made a proclamation for a thirty day period of national mourning. Congress did the same, and declared Washington's upcoming birthday, on February 22, 1800, a day of national mourning.

News had reached Philadelphia by December 19. Philadelphia newspapers that day and the following day announced Washington's death. Many newspapers would reprint verbatim the initial news account in the Alexandria Gazette.

On December 21, the Alexandria Gazette reported a lengthier account of Washington's passing, another news item that would be reprinted widely.

According to McGroarty, news had reached Boston by December 24, first reported in the The Massachusetts Federalist on Christmas morning, and also took the occasion to re-print Washington's Farewell Address in its entirety. By December 30, news had reached Portland, Maine.

Southward, the major newspaper in Georgia was the Georgia Gazette, operating in Savannah. They reported the story on January 9, 1800, taken from the first news report in Philadelphia on December 19. This likely was received by ship (though given the time in between, that ship likely did not leave Philadelphia that day, unless it made several stops along the way to Savannah).

In those days, newspapers had subscriptions to other newspapers. There wasn't anything like the Associated Press at the time. Instead, editors/publishers/printers scanned out-of-town newspapers for interesting items. Any major news event like this would be printed as soon as news was received. Something this major was likely distributed as an "extra" broadside even before the next morning (or evening) edition. News would have spread as quickly as news was received. The only constraint was the time to travel by ship or by stagecoach.

Given that Savannah heard the news by January 9, most Georgians probably heard the news within a couple weeks of that date, and probably sooner, given that "a poor farmer in Georgia" more than likely still lived within a week or two's travel time to the coast in 1800. News would have traveled up the Savannah River, and any adjacent land within the next days. Atlanta wasn't founded until a couple decades later; most of the interior of the state was still Native American territory until the 1820s. Athens existed, but wasn't big enough to be designated a "town" until 1806, and anyway, is close enough to the Savannah River, that it would likely have taken less than two weeks for news to reach the town.

Georgians, and Americans in general, who lived further out in the country would have heard the news later, probably the next time they went into town within a couple weeks of January 9.

By February 22, 1800, the day of national mourning as declared by Congress, the entire nation would have heard, except for those living in the most remote parts of the country.

To answer your question about how people learned the news, the majority would have heard it through the newspaper and through word of mouth, or often, a combination of the two. City-dwellers likely had a subscription and would have read it in the morning or evening newspaper. But with a story of such national interest, people were probably telling their friends and neighbors just as soon as it hit town, maybe carrying a newspaper with them. Or possibly with a specially published broadside by the town newspaper/printer to break the news a day or even few hours early. Either way, newspapers and word of mouth would be the ultimate source of the news of Washington's death for virtually everybody. The newspapers themselves would have learned it through their subscriptions to other newspapers. Depending on location, those could have been transmitted by stagecoach overland, by ship down the coast, and by riverboat into the interior of the country, and sometimes a combination of all three.

For more information on the national reaction and how the news spread, seek out the book The Long Farewell: Americans Mourn the Death of George Washington by Gerald E. Kahler, published by the University of Virginia Press in 2008.