Mammoth Cave National Park is the obvious day trip from Bowling Green — about 40 minutes north on I-65. The cave system is the longest known on earth, with over 400 miles of explored passages. Tours range from the easy Historic Tour (an hour and a half through the cave's grand passages and former tuberculosis hospital) to genuinely strenuous wild caving expeditions. Book tours in advance through the National Park Service website, especially in summer when they sell out days ahead.
Nashville, an hour south on I-65, is an easy half-day or full-day excursion. The city has transformed into one of the most visited in the American South, with a walkable downtown, live music on every block of Broadway, world-class museums (Country Music Hall of Fame, Tennessee State Museum), and excellent food.
Shaker Village of Pleasant Hill, about an hour northeast near Harrodsburg, is one of the best-preserved Shaker settlements in America: 34 original 19th-century buildings on 3,000 acres, with demonstrations of Shaker crafts, farm life, and architecture. Less well-known than Mammoth Cave but genuinely fascinating.