The key to obtaining quality public records is reporting. Before you can get what you need, you need to know what you need. For example: if you’re trying to follow up on a civil case, knowing the names of the litigants is helpful. For criminal cases, knowing case numbers can get you out of a jam. So you might have to report a bit and gather selectors and identifiers before you begin searching through your actual records. If you can, try and obtain the jurisdiction, geographic location of the court proceeding, and the name of a litigant before searching.
Your status as a student is your friend. That’s because you have access to LexisNexis’s databases and a slew of other online records repositories that the general public would have to pay for. http://libguides.usc.edu/mainsearch is a place to explore what records you have access to. For court proceedings in Los Angeles County, start here: https://www.lacourt.org/paonlineservices/civilindex/cipublicmain.aspx? You can search by name. Summaries of closed civil cases can be found here: https://www.lacourt.org/paonlineservices/civilindex/cipublicmain.aspx. A listing of all defendants in ongoing criminal cases can be found here: https://www.lacourt.org/paonlineservices/criminalindex/publicmain.aspx