I do a daily podcast using my phone and publish it on a wordpress blog. It's as simple as turning on the "voice notes" feature on my mobile, recording about 20 minutes worth, translating it to MP3 for distribution, and putting it onto my server with the correct links. I taught a course using that technique last July. It was an interesting experience from an instructional perspective.
You can actually make it much simpler than that by using tumblr and the dialup option to post audio to your tumblr site by calling a phone number, punching in a PIN, and then "leave a message" which then shows up online as an mp3 audio file.
The interesting questions involve whether or not this is really a podcast (that is, a series of files delivered via RSS enclosure) or just an audio or video file that your teacher wants you to turn in for a grade as a one-off that nobody will every hear. In the former case, you've got some issues of content and sequencing to address. Who is the audience? What content will you cover in each episode? How long will each episode be? That kind of stuff.
Good luck and I hope it works out well for you.