What's the best way to save voicemails from my home phone onto something permanent?

These days you can access them over the internet through your phone carrier, and save to disk. Then you can always have a digital copy.

Another way is to get one of those tape cassette recorders with the small tapes, a good quality one, record them to the tape, and save them that way. I did that for a phone message I needed to save from my phone answering machine before. This way you can erase them from your machine with ease. Just make sure and listen to the tape to be sure it sounds good on the tape, before deleting the originals from your phone.

I wanted to do the same thing for my Mom’s last few vms and knew several ways to do it. But for me, the simplest way was simply to record them to my computer using its internal mic. I then created a desktop folder and stored the audio files in it.

Are the vmails on a CD, or stored on some other type of medium?

Can you call in remotely and listen to your messages?

You could call in with a cell phone and use a call recording app to capture the audio.

Or, similar to @Pachy‘s idea, but swapping in a phone for the PC – use your phone like a tape recorder and simply hold the mic to the home phone earpiece