Grants are not free money. It takes a lot of work to get them, for one thing, and getting or not getting grants can have a lasting impact on your academic career. Small travel grants during your M.A. and Ph.D. coursework will help you get the big grants for your dissertation research and writing, and those dissertation grants will make you a stronger candidate on the academic job market. After you graduate, there are a number of early career opportunities available only in the first three or so years out of your Ph.D. Make a grants timeline for yourself as early as you can, and identify grants and deadlines for the different phases of your career well in advance.
Grants for Graduate Students Workshop Handout [PDF] provides an overview of the grant-seeking process
The best way to find grant opportunities is to stay abreast of Requests for Proposals (RFPs) in your field. Subscribe to listserves maintained by interest groups at UT and by professional organizations in your discipline. Talk to your professors and fellow graduate students about grant opportunities they've applied for (or check out their CVs). You can also check the websites below frequently (or subscribe to their email notifications) to explore new opportunities and stay up-to-date on trends in the kinds of projects funders are supporting.