In the summer before a student's final year of the PhD program they should begin thinking about applying for their first positions after graduate school.
The Graduate Studies Committee usually does a few-hour workshop for graduating students on writing postdoctoral position applications in the early summer. The first postdoc applications are generally due in September, with additional deadlines throughout the fall and winter months. The first notifications come around February, with some offers of postdoctoral researcher positions not being finalized until May. Students should update their CVs and webpages (including links to papers) and start thinking about their research proposal (usually 3 pages) in mid-summer.
Some resources for graduate students looking for postdoctoral, staff, and faculty positions include
Students interested in careers outside the postdoctoral researcher track are encouraged to talk to the Graduate Studies Committee Chair and/or their advisor. The Astronomy Department maintains a list of alumni contact information, interests, and experience that is a valuable resource for students interested in careers outside academia.