On your private tour of Athens you'll delight in the Plaka, a lovely pedestrian-only labyrinth of alleyways adorned with restaurants, jewelry stores, tourist shops, tavernas, ancient ruins, and Byzantine churches. It connects with the ancient Agora and has been used for over 2,500 years as the gathering and shopping area. It definitely is a shoppers delight.
At one time, it was the nightclub district, but most of these establishments closed down when the government outlawed amplified music. Arguably the most fun neighborhood in central Athens.
Note: Gold jewelry in Greece is inexpensive, not because the price of gold is any cheaper but because labor is less costly. There are bargains for leather goods as well.
FYI: Most of the streets have been closed to automobile traffic, though you should still keep a watchful eye for a speeding motorcycle or delivery truck.