If you don’t have iTunes installed, and you don’t want to install iTunes, then search for the podcast you want using google (say, “NPR car talk podcast”). You should be able to easily find the page from the original source offering a podcast.
For the search terms I quoted above, this was the first result.
You can bookmark this site, and return to it each day (or week, or however often the show is on) for a new podcast.
Different shows on different sites are offered in different ways. Sometimes, you can only download through iTunes. Sometimes, you can stream directly from the site. Sometimes, you can download the podcast onto your computer directly from the site and play it from there. In this case, if you want to stream Car Talk, you can only do so using iTunes.
I don’t like to stream, and using iTunes is complicated for me because I keep music on an external hard drive (this is not relevant to your issue), so I usually choose to download the podcast to play later.
To do this, find the button that says “Download” or find the button that says what type of file the podcast is in. In this case, there’s a button that says “Download MP3/4.” Click on it, and Windows will ask you if you want to “Open” it or “Save” it. Click on the down arrow beside “Save” and then choose the “Save As” option – this will allow you to navigate to the desktop or wherever you like, and save the file in that location. I think that if you just click “Save”, it will go to a temporary file where it might be hard to find.
Once that is done, you can go to wherever you saved the file on your computer, and play it using whatever media player you choose. Let us know if you have trouble with this – normally these files are associated with an Apple program called Quick Time, which you might not have installed if you don’t have iTunes. But there are ways around this. Try playing it first and see if it works.
Alternatively, if you have a different browser than I do, the “Save As” option might not be as obvious. In that case, it might be necessary for you to right-click and select “Save target as” which gives the same result.