Spring and autumn are the easy answers — warm but not muggy, with mild evenings for the patios and outdoor concerts. The big-event calendar is built around three things: the Indy 500 over Memorial Day weekend (the city's biggest event by an enormous margin, with the 500 Festival running for the two weeks leading up to it), the Brickyard NASCAR race in August, and the Indiana State Fair in mid-August.
Summer is hot and humid; the Children's Museum and the State Museum become heat refuges. Winter is grey and cold but the Winterlights display at Newfields and the Christmas decoration on Monument Circle are highlights. Race weekend rates triple — book a year ahead if attending.