Some universities—especially those that end with an Institute of Technology—have undergraduate degrees in nuclear science. But if the school you go to doesn't offer it, then physics, chemistry, or engineering are good ways to get started.
Unless your ambition is to be a nuclear science technician (basically a power plant operator, for which you'd normally need a high school diploma and a two-year degree), it'd behoove you to further your education with a master's degree or a Ph.D.
These degree programs let you focus on specific areas of nuclear science such as radiation protection, waste management, medical physics, or chemistry—specialties that will give you special skills and special opportunities, and an extra-special increase in income.
In addition to your nuclear knowledge, 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 these things). It's the state's way of saying yes, you do get to handle nuclear material; just don't spend it all in one place (source).