Library Main Page > Research Databases > BNA Electronic Publications > browse the list to United States Law Week. You can access this directly on campus; from off-campus be ready with your UConn netid and password.You will land on the US Law Week page. This page is a prime source for national legal news. BNA takes pride in source-checking its stories. They have a regular staff of reporters.
For the USSC page, use the hyperlink in the left column - Supreme Court Today Home Page. Your landing page (Most Recent Action) will be deceptively simple and spare but quite functional. Browse down it to find links to summaries of recently argued cases and recently entered orders (granting or denying certiorari) . There is a search function that will allow you to find previous cases.
Use the navigational links in the left hand column to get to the Court's Journal, the Argument Schedule, and the Status of cases. You can search for the text of Opinions by case name, court term and topic. And, use the Argument & Review Stories to get the summary of all of the important activity in the case including an analysis of what happened at oral argument and the issues raised by the briefs.
Back at the U.S. Law Week site there is more than just current news: the Research Tools function that includes human generated indexing; a From The Editors feature that puts the past term in perspective: the Circuit Splits feature where federal Circuit courts of appeal splits are noted.
So, is this the best answer on how to stay current with the USSC? Just remember that this is a fee service. All of this great, reliable information comes with a price tag. And, unless you are a paid subscriber (like UConn Law is) you can't get updates delivered automatically by email.
If you want free, you might want to try the newcomer - SCOTUSBlog.