Dave,
I hope you'll love Kiawah as much as I do. We visit 2 or 3 times every year. I have been fortunate to play both River and Cassique a few times. Both are wonderful, first class private club experiences, with spectacular service, beautiful surroundings and immaculate conditions. As for the golf, I prefer Cassique. Technically, it's much less "natural" than River (I've read somewhere how much dirt was moved to build Cassique, it's mind boggling), but it feels natural - like a true links course. I found it more a more interesting round. Take a caddy (may be required) as walking the routing can be confusing for the first timer -- in a few locations several tees are in the same vicinity.
River is a really nice course, and I love the short par 4 on the front (no. 4 or 5?). It is a bit "typical Fazio," in that there's lots of eye candy, but I found it to be a strong course which presented appropriate options for both the low and high handicapper. The River Course is more punishing if you're a bit wayward with the driver.
If you have access, you really can't go wrong with either course. I find them to be the second and third best courses on the island (after Ocean) by a long shot.
The clubhouse at Cassique is among the most beautiful I've ever seen. The clubhouse at The River Course burned to the ground last year. I just returned from the island, and while they've started reconstruction, they've a long way to go.
As for the public options, The Ocean Course is really a world class round of golf. Be warned, it is very difficult, so pick the proper tees. The tariff to play is quite high, north of $250 plus $60 tip for the caddy, but it is worth it. Walking is required before noon (love that). Do yourself a favor and have a late afternoon cocktail on the porch of the clubhouse.
As for the remainder of the courses, I'm a bit of a contrarian on GCA in that I really dislike the Nicklaus course, Turtle Point. I would rank the courses on the island as follows: Ocean, Cassique, River, Osprey Point (Fazio), Cougar Point (Player), Turtle Point (Nicklaus) and Oak Point (not worth the 1 mile drive to get there -- really bad, even after the re-do). I have not played "off island," so I cannot comment on other options. Obviously, Yeamans Hall is worthwhile, but I find the drive in to Charleston to be more than I care to undertake while on vacation -- realistically about 45 minutes from behind the gates of Kiawah.
Good luck and have fun. Take or rent bikes -- it's a great way to see the island, and biking on the beach is great fun.