Kroto was educated at Bolton School and went to the University of Sheffield in 1958, where he obtained a first-class honours BSc degree in Chemistry (1961) and a PhD in Molecular Spectroscopy (1964). During his time at Sheffield he also was the art editor of Arrows – the university student magazine, played tennis for the university team (reaching the UAU finals twice) and was President of the Student Athletics Council (1963–64). Among other things such as making the first phosphaalkenes (compounds with carbon phosphorus double bonds), his doctoral studies included unpublished research on carbon suboxide, O=C=C=C=O, and this led to a general interest in molecules containing chains of carbon atoms with numerous multiple bonds. He started his work with an interest in organic chemistry, but when he learned about spectroscopy it inclined him towards quantum chemistry; he later developed an interest in astrochemistry.
After obtaining his PhD, Kroto spent two-years as a postdoctoral fellow in the molecular spectroscopy group of Gerhard Herzberg at the National Research Council in Ottawa, Canada, and the subsequent year (1966–1967) at Bell Laboratories in New Jersey carrying out Raman studies of liquid phase interactions and worked on quantum chemistry.