If you're looking for the most picturesque place on Kangaroo Island, Vivonne Bay is definitely a contender. You'll probably even recognise it from advertising material. It comprises a beautiful stretch of sand with two little hamlets at the back. The water is a beautiful blue and there's a jetty that juts out over it.
Two different points provide access to the bay. One of these involves taking Knofel Drive off South Coast Road to the main town behind the beach. It's best for access to the sand and there's also a river, but the town itself doesn't have too many facilities.
Another cluster of roads and buildings is located down near the western headland. For this area you will need to take the Jetty Road or Roswells Road turn-offs. There's no sand up this end but you can reach the jetty and walk along the headland. Other walking and biking trails through the bushland are signposted around Vivonne Bay.

I did the headland walk

I could barely see the beach but the view was still breathtaking
The bay isn't very close to any main towns (a good thing or a bad thing, depending on your perspective). However, being located on the southern side of the island, it is near a lot of the island's top attractions. To the west are
Hanson Bay Wildlife Sanctuary and
Kelly Hill Caves, and to the east is
Seal Bay Conservation Park and
Little Sahara.
If beautiful scenery is what draws you to Vivonne Bay, I recommend also visiting Pennington Bay, which I found quite pretty too, though its surf means it isn't ideal for swimming.
Stokes Bay is one of the better alternatives I found if you want to actually get into the water (it even has a secret entrance and a tidal pool).
Accommodation can be found at Vivonne Bay if you intend to stay for a while.