What's the best way to travel to Monemvasia and Mani from Corfu?
From Corfu, take a ferry to Kefalonia, and then another ferry to Patras.
Once you're in Patras, you'll need to find a rental agency that allows a one-way rental so you can drop the car off somewhere else (like Athens or Kalamata). You will likely have to pay a hefty drop-off fee for this option.
While the Mani Peninsula is definitely worth a few days of exploring with a rental car, you’d better be really interested to justify driving all the way to Monemvasia. It’s located on the exact opposite side of the Peloponnese from Patras, and the medieval town is basically just a completely uninhabited, open-air museum that you can easily see in 2 or 3 hours. (for a one-way trip of nearly 6 hours from Patras).