Nashville is called Music City because a radio show made it one. The Grand Ole Opry began in 1925 as a Saturday-night broadcast on the local station WSM, and it is now the longest-running radio broadcast in US history. Record labels followed the singers, studios followed the labels, and by the late 1950s a few blocks of old houses southwest of downtown, Music Row, were turning out a large share of the country hits in America. The industry is still there, and so is the habit of songwriters playing for each other in small rooms.
Three days is the right length. The good things are close together and most are short visits: a museum, a studio tour, a seat in a listening room. What takes time is the evenings, because Nashville does its best work after dark, and three of them let you do the loud one (Lower Broadway), the formal one (the Opry) and the quiet one (a songwriters' round at the Bluebird). Franklin, Lynchburg and Memphis are worth a trip, but they are separate trips, not a day to squeeze in here.
Day one is downtown and Studio B, day two goes east to the Hermitage and East Nashville and ends at the Opry, and day three covers the Parthenon, a whiskey tasting, 12South and the Bluebird. More on the country in the United States guide. Start on a Thursday: the Opry plays on Tuesdays, Fridays and Saturdays, so Friday night gives you a show, Arnold's Country Kitchen is a weekday lunch, and the Parthenon, the whiskey tour and the Bluebird are all open on the Saturday of day three.