Good question. I'll answer only to the United States, since I don't have a good grasp on an answer for other places in the world.

On May 1, 1857, the Washington Evening Star published the first weather forecast in the United States. This was the result of eight years of active work and decades of preparation by Americans across the country. Taking weather observations had been a common activity among science-minded people as early as the 17th century. Using new, accurate barometers made possible by the advances of the early modern era, they could measure air temperature and pressure, wind speed and direction, cloud cover and precipitation.

Everyone knows about Benjamin Franklin's fascination with the weather, but Thomas Jefferson was an avid weather-watcher, too. He bought his first thermometer while writing the Declaration of Independence and bought his first barometer a few days after signing that document. He noted that the high temperature in Philadelphia, Pennsylvania on July 4, 1776 was 76 degrees. George Washington also made regular weather observations. The last weather entry in his diary is on the day he died.

Through the late 18th and early 19th centuries, Americans charted the weather. In the 19th century, weather observing networks began to grow across the United States. Americans would take their observations and send them in to a central collecting point, which would collate the results and publish them. Americans could see where it was wet and where it was dry across the country. One of the pioneering efforts was made by the surgeons of the U.S. Army, who recorded observations reliably from the 1820s to the 1840s at posts throughout what was then the frontier. The Army Medical Department's ostensible goal was to determine what effects the weather had on disease.

The data collected by these observers wasn't "live," though. The mail took weeks or months to travel from state to state, which meant that the information was of historical value only. This changed with the invention of the telegraph. When Samuel Morse patented the practical telegraph in 1837, it became possible to send news of the weather faster than the weather itself.

It took time for this to happen. The telegraph spread rapidly -- almost everyone could see its value -- but charting the weather required an organized network of both observers and transmitters.

Joseph Henry, the first secretary of the Smithsonian Institution, had the foresight to see what was needed. In 1848, he began a project whose goal was to describe the climate of North America and learn about storms as they traveled across the country. At the time, it was not even known that in North America, storms predominantly travel from west to east.

Henry had already recruited volunteers -- teachers, farmers, ministers, lighthouse keepers -- across the country to take measurements that they then mailed to the Smithsonian on a monthly basis. In 1849, he recruited a smaller group of volunteers who telegraphed their daily observations to the Smithsonian. The Smithsonian supplied these volunteers (many of whom where employees or representatives of the telegraph companies themselves) with calibrated instruments, standard reporting forms and guidelines for collecting data.

The telegraph companies realized there was value in sending this data free of charge, and Henry used the data creatively. Starting in 1850, he began preparing a daily current national weather map in the Smithsonian castle in Washington, D.C. This map was later put on display in the castle's main hall, where it could be viewed by the public.

Newspapers and others took this information and distributed it. As people realized that the weather followed specific patterns -- notably from west to east -- they realized that not only could they observe the weather, they could predict what would come, based on what the weather was doing elsewhere.

There were problems with Henry's program, of course. Observations were based on local sun time, which meant they weren't taken simultaneously. The start of the American Civil War in 1860 disrupted the program for almost a decade, and the data was only as accurate as the observers were. If someone was sick or simply failed to take one of the four readings of the day, the information was missing. In 1870, President Ulysses Grant signed the first legislation creating a regularized weather service within the U.S. Army's Signal Service. This service grew and became the U.S. Weather Bureau (today the U.S. Weather Service) in 1890.