For a day trip, best things to see besides golf courses? Nice place for lunch? Preferably near water.
The golf museum has a lovely restaurant on the second floor with a beautiful view out over the golf course and out over the water. I enjoyed my meal there and the view.
I also had a blast playing the Himalayas course. This is the Ladies Putting Association Course and it is so much fun. You really don't need to be much of a golfer to do this. It is a bargain, just a few pounds depending on age. Take the time to play the course!
Walking on the beach is very pleasant as well.
The Martyr's Monument, next The Royal and Ancient clubhouse.
There is a wee university in town, founded in 1413. My brother went there for a year and says it is the model for Hogwarts.
There's the castle and the cathedral, both of which are ruins, but are close by one another, and well worth a visit. Also, the beach which Carol mentioned is the one which was used in the opening scenes of the Oscar winning "Chariots of Fire." Bring along a pair of knee length white shorts and a white crew neck t-shirt, and you can re-create the opening scene. Don't forget to run barefoot.
The opening scene is on YouTube. You can see the town in the background as the camera pans right. The "Carlton Hotel" is actually the R&A clubhouse.
In addition, there are some very nice shops and restaurants along St. Mary's Place/Market Street.