You will need to know what you are looking for when you start your search.
There are a few different types of general Graduate Assistantship positions: teaching assistants (TAs), research assistants (RAs), and graduate assistants (GAs).
After that, you want to check in with the contacts you have made in the graduate school office, the faculty of the department, and the students who currently attend the program.
You can also do a search on Linked In or on the departmental pages to see if the current GAs are listed for the program.
For example, if you are interested in an athletics administration assistantship, you could check out the NACDA website, and discover the listings they have for Graduate Assistantships related to collegiate athletic administration. Each industry and interest area will have different websites of value to your search.
Just like the individual programs, you need to start keeping track of application deadlines and materials needed for the assistantships.
GAs will function in several different capacities across the university; perhaps directly for the individual academic program or possibly within an administrative capacity somewhere else across the functions of the university (like Residence Life, Athletics, Enrollment, or others).
For this reason, you will want to check within the academic department of the program to which you are applying, as well as the institution as a whole when searching for an assistantship.
These networking contacts can let you know about available assistantship opportunities and how to stand out as a top candidate.
You can also use regular job search engines to find assistantships.