Spring (April and May) is the best time to visit Knoxville. The dogwood trees bloom across the city in April, which the annual Dogwood Arts Festival celebrates with outdoor exhibitions and events. Temperatures are mild, the Smoky Mountains are accessible, and the city is at its most photogenic.
Autumn (September through November) is the other strong season. The mountain colours in October are exceptional and Knoxville fills with energy from the University of Tennessee football season — which is either an attraction or a reason to avoid certain weekends, depending on your tolerance for orange-clad crowds of 100,000 people.
Summer is hot and humid, with temperatures regularly above 35°C in July and August. The Smoky Mountains offer cooler refuge, but popular trails and Gatlinburg itself get extremely crowded. Winter is mild by comparison with much of the US — snow is rare in Knoxville, though the mountains may have snow from December through March.
The Big Ears music festival in late March is worth timing a visit around if that sort of programming appeals.