Fill out a FAFSA and apply for federal aid. To receive any type of grant, you must first fill out this official form and try to qualify for a Pell Grant (the most widely distributed federal grant.) After applying, you will receive an "Estimated Family Contribution" (EFC.) This number will be very important when you are applying for other grants. Even if you do not end up qualifying for a Pell Grant (which is need-based), you will have to have a FASFA on file before you can qualify for any other grants.
Bring your completed form to your Financial Aid office. Let them know that you are actively looking for other grant opportunities within the school. You will probably need to fill out some documents for your school to make sure you fit their standards for other grants. Make sure to specify whether you want to apply for need-based, academic, or research grants.
Look into research projects in your area of academic study. Talk to professors and department heads and see if you can become a grant-funded research assistant. This is a very popular way that those who do not qualify for need-based aid become grant recipients. However, becoming a research assistant is not an easy process, and you will need relevant experience and a polished resume in order to be considered for the position.