Federico was born in Genoa on October 25, 1997. He started his career with Settignanese in Coverciano and joined Fiorentina when he was 10 years old. After moving through the Viola youth ranks and making his senior debut on August 20, 2016, at the age of 18, he would return to Coverciano a few years later as a member of the Italy setup. He quickly became a mainstay in the team, appearing in 137 Serie A games for Fiorentina over the course of the following four seasons.

With his lightning speed, Chiesa was able to outdistance opponents right away. His first Serie A season (2016–17) saw him complete one of the highest numbers of dribbles in the league: 533.