We decided to visit Beijing in the cold of winter to avoid the crowds of tourists, and be able to enjoy the bright blue skies which only tend to come with freezing temperatures. The capital of China has a unique atmosphere in winter, with buzzing activity in the hutongs despite the cold, and with spectacular clean landscapes washed by snow or ice. From Singapore, Jetstar has a convenient direct flight, and Beijing can be enjoyed in 3-5 days before considering exploring the region around it. I would recommend finding a guesthouse in the Hutongs, or in the neighborhood o QianMian. The MRT system is very good, the bus however requires you to recognize Chinese characters to establish your route... not ideal! And as far as taxis are concerned... forget about them. Its strangely very difficult for tourists to board taxis because drivers aren't too keen on wasting time trying to understand a destination.
By strolling in the Hutongs, you will see many odd shops like this ambulant library, and many more street-food stalls selling about anything you might want! In winter, the mix of steam from people's breath or from the numerous sources of heat gives a unique atmosphere to the city!
Obviously, one of the main sites of the city center, is the fascinating forbidden city. Today a major touristic site, its possible to visit most of the palaces, gardens, halls, temples and pagodas which rest within the wall & moat. The towers guarding each doors and each corners of the fortification are impressive, and again, the winter's frozen moat gives the site a unique atmosphere! Quite a few soldiers patrol both within the forbidden city as well as outside, near the tiananmen square. In winter, these soldiers wear the impressive cold-weather uniform. Although the Forbidden City is usually very crowded, going early morning and in winter does allow you some space, and the fact that most tourists are Chinese people sort of keeps the atmosphere authentic!
Beijing has many more sights, and museums. Just opposite the forbidden city, you can access the huge beihai park with its beautiful lake (recommended in the morning to see all the people doing TaiQi or KungFu by the water, or by night to enjoy the illuminations), and the Jingshan park which offers stunning view on the city and on the forbidden city from the heights.
Further south, the Temple of Heaven and surrounding gardens is a fantastic sight not to be missed. Its actually very enjoyable to walk there from Qian Mian because the route crosses many typical streets. Further west from the temple of heaven, the NiuJie Mosque is a peculiar sight because it is a quiet mosque with the typical arabic calligraphy, but in the Chinese pagoda style. Its worth a visit also to be able to stroll in the Muslim neighborhood and its great mutton skewers and noodles!
Visiting the great wall usually takes either a day trip of a half-day trip. My recommendation is not to go to badaling, because of the horde of the tourists which literally make you queue-up on the wall! The scenery is impressive, but the crowd is slightly overwhelming. The Huanghuacheng section of the wall is more remote, and is the easiest to get to from Beijing while avoiding the crowds. There is a lake at the bottom and the site is amazing!
Another great get-away from the city center is the summer palace. The palace is accessible in MRT, and the park is huge with a lot of halls, lakes, temples, pagodas and so on. You would need at least half a day if not a day to enjoy the place! In winter, the frozen lake is spectacular!