Research opportunities for graduate students abound. In addition to working directly with a Physics Department faculty member, graduate students also conduct research with faculty in Astronomy & Astrophysics, Chemistry, Geophysical Sciences, Biological Sciences, Molecular Engineering, as well as at Fermilab, Argonne, CERN, and many other facilities.
In many cases, graduate students can get involved in research immediately upon, and sometime before, beginning their first quarter in the program. The best way for graduate students to get involved in the research of his or her choice is to communicate with the faculty members working in their area(s) of interest to find out how to best proceed based on their specific situations. A great way to explore an experimental research group early-on is to satisfy the experimental physics requirement by doing a year-long project with that group.
You can get a taste for some of the research performed by graduate students by reading these profiles of current graduate students.