Some universities—such as M.I.T.—have undergraduate degrees in nuclear science. But if the school you go to doesn't offer it, physics, chemistry, or engineering should get you started.
Unless your ambition is to be a nuclear science technician (basically a power plant operator, where you'd normally need a high school diploma and a two-year degree), it'd behoove you to further your education and even go on to obtain a master's degree or a PhD. Some of the most successful nuclear scientists working in the field of engineering, physics, chemistry, or medicine obtain their PhDs, and even go on to take post-graduate classes to keep on learning. These degree programs let you focus on specific areas of nuclear science such as radiation protection, waste management, or medical physics or chemistry. As well, most states require that nuclear engineers obtain a license that follows the guidelines of the National Council of Examiners Engineering and Surveying (NCEES, in case you need an acronym to help you remember).
And if you want to work in education at the university level, a PhD is oftentimes a requirement.
But it's not necessary to get a PhD or a master's; having a broad education with a focus on the sciences is a good way to get started.