Teaching on undergraduate courses is typically not available to postdocs. However to gain some teaching experience you can propose a short (normally 8-lecture) specialist graduate course held at IoA on a subject close to your research area.
If interested, please contact Paul Hewett (Director of Postgraduate Education) to see what slots are available. If you want to see the current programme, look here.
Postdocs are often offered (paid) opportunities to do supervisions for undergraduate and graduate courses . For courses run within the IoA email requests for volunteers to take supervisions are often sent out at the start of the academic year. You can find out more about the undergaduate lecture series and who you can ask about supervisions at http://www.ast.cam.ac.uk/students/undergrad.
If you undertake undergraduate supervisions you have to complete a training course (details here), which is generally held in October.
Postdocs are canvassed enthusiastically to provide projects for the Part III Masters programme. Requests for projects are generally sent at the start of summer with several reminders sent over the following months until around the start of September. After this time you have to wait until the following year.
This is one of the best ways to get experience supervising students in a research context at Cambridge.
Details and the current project booklet can be found at http://www.ast.cam.ac.uk/students/undergrad/part_iii.
For supervisions through the colleges (including supervising a part 3 student) you need to claim the hours and so the pay. For all colleges this is done through CamCORS, a central claiming system: CamCORS.
You log in via Raven, and create and submit reports per student per term. The first time is a bit strange to navigate; contact committee member Emily Sandford (es835@cam.ac.uk) if you need CamCORS guidance.
You will need the CRSID of each student, which is either the first section of their email, or you can look them up in the Cambridge University Directory: UC Person Lookup.
As of 2017 for Part III students you can claim for 8 hours over the whole project (confirm this when you are allocated a student). You submit reports for each term shortly summarising their progress.