I go down to Hilton Head twice a year cuz my parents own a house down there. I have played just about every single public course on the island, and some off the island. Here's what I thought:
In Sea Pines:
Harbour Town - the best on the island (if you have $250 to spend on the greens fees then do it)
Ocean Course - the island's first course. it's very good. tough and fun with lots of water. 15th green provides view of beach/ocean. I recommend it.
Heron Point - new by Pete Dye. tough, but not too memorable. however, if you're a fan of Dye, I'd play it.
In Palmetto Dunes:
Robert Trent Jones Course - my second favorite. big, deep bunkers and huge greens. lots of water. 10th green is literally right next to the beach/ocean. I highly recommend it.
George Fazio Course - haven't played it, but I have heard it's one of the island's best
Arthur Hills Course - haven't played it
In Port Royal:
Robber's Row - pretty good, but not very tough or memorable.
Planter's Row - average. short, and not too challenging.
Barony Course - haven't played it
In Indigo Run:
Golden Bear at Indigo Run - I don't remember much from this course, I just remember that I didn't like it. Not one of Jack Nicklaus' best designs.
In Hilton Head Plantation:
Country Club of Hilton Head (semi-private) - very good. tough track, with no room for errant shots, as OB is very close to the fairway edges. some reachable Par 5's, and lots of tough Par 3's.
Oyster Reef - average. not a memorable course.
Shipyard:
27 holes - A LOT of water here. courses are not very long, but they require well-placed shots to score well. not too tough, but it is a pretty fun layout.
Off Island:
Old South Golf Links - challenging, fun, a lot of water. Lots of rolling terrain. 8th hole is a par 3 with Island green. I recommend it.
In short, the courses to play are:
Harbour Town (if you have the money)
Robert Trent Jones at Palmetto Dunes
George Fazio Course (it's excellent from what I've heard)
Ocean Course
Country Club of Hilton Head (remember, it's open to the public)
Old South Golf Links
Courses to put on your "back-up list":
Heron Point
all of Shipyard
Robber's Row
Arthur Hills at Palmetto Dunes
Courses to Avoid:
Golden Bear at Indigo Run
Planter's Row
Barony Course
Oyster Reef