Your assignment from Shaftesbury gives ample evidence that he was influenced by neoplatonism. Since he also presents a couple of views which came to prominence in the period following this work (which was published early in the 18th century), he provides a link between the long tradition of neoplatonist aesthetics and new ideas that we will find in Hume and Kant.
• A distinction between beauty and usefulness, and between the appreciation of beauty and interest. For this, see HK 246-248 and HK 258-259. Notice the way an appreciation of the wilder aspects of beauty (as discussed in the initial exchange, HK 244-246) is tied to both a higher or truer sort of beauty that may not be noticed immediately and to the distinction between appreciation and possession.
• The idea of a natural sense of beauty (and an analogous moral sense). This can be seen as the chief topic in one way or another from HK 253 to the end (HK 261). Note especially how Shaftesbury deals with the problem of variation in taste among people (beginning on HK 255), an issue that will be important in both Hume and Kant.
Each of these views raises an issue that we may spend some time discussing:
• Is there a fundamental difference between beauty and usefulness or between the appreciation of beauty on the one hand and the possession or consumption of beautiful objects on the other hand? (For a contrasting view, you might think of the way both Plato and Aristotle link value with use.)
• Do we have an innate or natural sense of beauty? (Shaftesbury is unlikely to have suggested this casually. John Locke’s criticisms of innate ideas would have been well known publicly and probably even more so to Shaftesbury since Locke was an employee and close associate of Shaftesbury’s grandfather and had even supervised Shaftesbury’s education for a time.)