Take a trip on the Batong line if you really want to 'go native'
Outside of the main tourist attractions things are really much of a muchness, and the city is becoming more and more homogeneous.
The hutongs are neither here nor there imo- you have a choice of the recently renovated areas around Houhai that are full of hipster boutiques and touristy shops. More like a themed shopping mall rather than a cultural experience.
Or there are the traditional hutongs where 'those of small means' actually work and live. These are pretty grim and grotty in general and not in an interesting or romantic sense.
Chaoyang Park is great and well worth a half day. A few other good local parks are Jiangfu and Side which are smaller but a lot more 'local'. The Birds Nest and Olympic Park also worth a visit if only to see the 'white elephantness' of it all.
The China Nationalities Museum (open air) is also highly recommend, that is just to the south of Olympic Park.
You will most likely get a native experience during eating, so just wander into any restaurant or pull up a stool at the nearest Chuanr place. There are quite a few pop up outdoor BBQ places around town and these usually have a great atmosphere with lots of Beijing Bikini's on show!
A day at a hot spring like Jiahua is also worth the one off experience.