If you want to see the quiet, rural side of Hong Kong (which tourists seem to ignore), I have two suggestions:
When you go to Ngong Ping (with the giant Bhudda), take the MTR to Tung Chung, but don't take the cable car. Look for a bus to Shek Mun Kap, to the beginning of a cement trail that leads all the way up the mountain to Ngong Ping. Bring plenty of water and snacks, since there are no facilities on the way up. The walk will take about 2 hours.
Lamma Island might be another place to visit, as it's basically untouched by modern civilization (save for a massive ugly gas power station, but it's easy to ignore it). Lots of westerners who live like hippies, and the seafood restaurants at Sok Kwu Wan are a hidden secret. There are ferries from Aberdeen and Central.
Be aware of the air quality, as many days have significant smog.
If there's one place you should visit in Beijing, which is not obvious to foreign tourists, it's Panjiayuan with its huge flea market. All sorts of historic objects are sold there, and you *must* bargain. Some hawkers from Tibet and Xinjiang are rather aggressive at selling stuff, so beware. Don't buy anything over 50 years old, because of cultural export restrictions.
Enjoy your trip.