I lived in Jacksonville for about a year, and there is a lot of golf. It ranges from quite expensive (TPC) to very cheap (Hyde Park). I didn't play much in St Augustine, but I would strongly suggest avoiding Royal St Augustine. They used to have a quarterly golf paper that had all the courses in the area, and the deals they were offering. I would play all over the place at really good rates.
At the World Golf Hall of Fame, the King and Bear is much better than Slammer and Squire.
There is a group of courses close to I-95 between St Augustine and Jacksonville that are easy to get to and good places to play regularly. South Hampton, St John's G&CC (careful there are two St John's in the area), and Cimmerone (sp?). They are all within a couple of miles of each other. All are modern housing development courses, and not spectacular examples of architecture, but all had good rates (avoid peak hours), and were fun to play somewhat regularly. Cimmerone's conditions were somewhat of a crap shoot, but the others were always in good shape. These might be good options from St Augustine.
My favorite in Jacksonville by a mile is Timuquana. It's an old Ross south of town, and is amazing. I wish I could play it everyday.
Eagle Harbor and Fleming Island are near each other, and decent everyday courses. Eagle Landing is a bit further, but it was a place I played often. Bent Creek is a good muni, but is also a bit of drive.
There are a bunch of good choices in Ponte Vedra, but most are private or expensive. The Ocean Course at the Ponte Vedra Inn is good. Sawgrass Country Club is solid, and very difficult. Marsh Landing is supposed to be good, but I haven't played it. Same goes for the Plantation Club or something like that.
The UNF Practice facility (open to public) was a great place to work on your game.