The NSF Graduate Research Fellowship Program is one popular and prestigious fellowship that can be applied for as a senior undergraduate (in addition to some of the others listed in point 1). You can also find other fellowships that you can apply for on the University of Illinois Fellowship Finder.
Having external fellowship support can open many doors – with a fellowship you become a ‘free agent’ that can work with almost any advisor that is willing to mentor you (because your advisor no longer needs to assume (full) financial support for your studies). Participating in undergraduate research and doing well in classes will also strengthen your fellowship application. For example, fellowship applications usually require a research statement which will be easier to write if you have research experience. You may also be able to get help with the statements from professors you’ve worked with in the past.
Additionally, completing fellowship applications will also prepare you for applying to graduate schools because most of the materials you need for the fellowship application (e.g., personal and research statements) will also be required for your application to most schools.