Your department is one of the best places for advice and guidance in considering fellowships and grants, along with your faculty advisor. The Department, particularly the Student Affairs Academic Administrator, has wide experience with many sources of funding, both general and specific to the discipline. Departments often have travel funds and small grants available to students, and some have an array of Departmental dissertation research and write-up support. The Departmental web site is a good place to begin.
Associate Dean of Students Kelly Pollock is available to advise you on the trajectory of fellowships in your academic career, and she can also assist you in writing your fellowship applications.
Graduate Student Affairs
Advisors in the Office of Graduate Student Affairs are happy to help identify fellowships for which you are eligible, and to help you with the administrative aspects of applying for fellowships. Additionally, they can help edit your essay(s) to make your application as competitive as possible. GSA also administers external fellowships, such as the Fulbright.