There are approximately 10-12 new PhD students in each incoming class. Students typically have funding for 3.5 years, and so their degrees are also either 3 or 4 years long. Each PhD student is affiliated with a college, and some colleges provide extra funding (e.g., for travel).
Each student has two official supervisors. All postdocs can help to supervise students, and senior postdocs (those on long-term Fellowships) can be an official supervisor. The students spend the first month or two determining what project they'd like to pursue for their PhD; if you have an idea for a PhD project then speak with your faculty contact about potentially taking on a student!
The Part III is the fourth year of a Cambridge degree. It is optional (i.e., students can leave after their 3rd year with a degree) and is approximately equivalent to a Masters program. The Part III consists of a significant amount of coursework and a research project.
The Part III research projects are often proposed and supervised by IoA postdocs (always with an official faculty contact, but they may play zero role in the project). Supervising a Part III project is 'supposed' to take 8 hours total from October to May, but in practice many postdocs spend much more time. The projects vary considerably in difficulty, scope, and reward; some will lead to publications, others will be a pure academic exercise. Vasily Belokurov (office H20) and Mahsa Zohhadi (H4) administrate the research projects and will solicit project ideas from IoA members in late July or early August; contact him if you're interested in supervising a project.
The Cambridge undergraduate course consists of 'Part 1' which lasts for two years and does not require students to 'specialize' yet, and 'Part 2' in their third year, at which point they'll choose to do Astrophysics, Physics, etc. Each undergraduate class consists of lectures (typically given by faculty) and 'supervisions', or small group (2-3 students) tutorial sections. Many of the IoA PhD students do supervisions, but some of the postdocs have also done them. There's a small monetary compensation for the effort.
The IoA has a summer research scheme aimed at students who have just completed their Part 1. These students work for 8 weeks over the summer and are typically highly motivated (there are a select number of spots, and they are paid). You will receive an email during the Lent term (February or March) soliciting project ideas. One difficulty of summer students is that we are often gone for part of the summer, but if scheduling can work out then this can be a good way to get experience in supervising a full time research student.