Hey there, great to hear about your trip.
The Mid Autumn Festival also known as the moon festival is held on the fifteenth day of the eighth moon and is one of the most popular festivals celebrated in Beijing.
I have listed below some of the best places to enjoy this festival and have also added some great events you can catch during the festival.

(Source: Beijingfeb06CNY 130)
Here are my recommendations:

Shogun (Source: Bill S)
A temple fair is held at Wanping Town east of Lugou Bridge each year to celebrate the Mid-Autumn Festival. It is a traditional and cultural extravaganza featuring all kinds of Chinese folk art. There are lion and dragon dances and demonstrations of traditional arts and crafts. Great way to understand the culture and lifestyle of the locals.

(Source: Wonderlane)
Location: 15 km (10 miles) southwest of Tiananmen Square, Fengtai District.
It is customary for locals to go to the Summer Palace to see the osmanthus flowers during the Mid-Autumn Festival.That said, the beauty and serenity of this place is enough to draw the attention of any wandering travellers.From the delicate art work ,to the magnificent architecture,Summer Palace is a must visit any day,any time of the year.

The Marble Boat @ Summer Palace (Source: Kevin Poh)
A less known attraction in the Summer Palace, the 72 ancient osmanthus trees make their contributions to the beauty of the imperial garden by their sweet flowers in fall. Family and friends can opt for a walk among the osmanthus trees, or take a boat ride along the lakes to enjoy the lakeside scenery. Boat rides during the Mid-Autumn Festival are very popular and are usually short of demand, so booking in advance is recommended.
(Source: Beijing Mid-Autumn Festival, Chinese Culture, China Highlights)
Location: 19 Xin-jian-gong-men Rd, Haidian District Beijing 100084
Tel: 010 6288 1144
Admission: RMB 50
For timings, see here.
Yangtaishan is another popular place known for its full moon during the Mid-Autumn Festival. The natural beauty of this place is breathtaking and one can take a walk along the tree-lined path or go fishing or set up campfire by the Sun and Moon lake.

(Source: Bill Song)
Location: 30 km (20 miles) northwest of Tian’anmen, in Haidian District.
Transportation: Take bus 346 at the Forbidden City and get off at North Anhe Station
A culinary experience like no other ,CCTV Tower at a height of 385 metres ,serves and amazing buffet of continental as well as traditional Chinese and Japanese cuisine at the one of a kind revolving restaurant in Beijing.The restaurant sets up telescopes in each direction,so that the guest can enjoy their food as well the festivities around.

(Source: Ryan Raffa)
Location: West Third Ringroad Central, Haidian District.
Call:6847-5552
Timings:11am-2pm, 5pm-7:30pm
Price : Y200-Y299
Here are some great events to not to miss during the festival:

(Source: Avery)
Since you have mentioned that you love food, I found this great Food festival for you.
Enjoy a great feast with along with live folk music and moon watching. Traditional Chinese cuisine will be served and I am sure you will love it.
Here are the details:
The feast will be held on the 12th of September and it starts from 5pm.
Timings:Live Folk Music & Snacks from 17:00, Casual Buffet Dinner Served from 18:00
Price: Adults $31
For more information and booking, call at Tel: 6162-6506

Beijing (Source: Bill Holler)
For more information on food festivals and where to eat during the festival click here
How about going for a concert? The Beijing TV Mid-autumn night performance are epic and adds a whole new flavour to the festivities.The showcase is usually held at the Marco Polo Bridge on September 12th.

(Source: Kathy Zhuang)
The theme for this performance will be Happiness on Mid-autumn with Lugouxiaoyue, bright moon seen at the Marco Polo Bridge).
The whole performance lasts for 3 hours at Marco polo Bridge, and during the performance, some scenes at Beijing TV, the Bird's Nest, Jingshan Mountain as subordinate courts are shown to the audience. It is a performance of dances and songs about moon and the Mid-autumn. With the background of the historic Marco Polo Bridge and the modern theatrics, it will surely be a wonderful night.
(Source: TV Performances Special for Mid-autumn Night)
Location:Marco Polo Bridge 19 Xin-jian-gong-men Rd, Haidian District Beijing 100084
The show starts from 7:40pm.
Beihai Park was the favorite place for the emperors and kings to watch the moon during the Mid-Autumn Festival. You can take a cruise on the lake to enjoy the night scenery. A variety of boats are available during the festival and the park remains open until 9pm.
Beihai Park, Beijing China (Source: Dmitry P)
A review:
(Source: TripAdvisor)
Location: northwest of the Forbidden City, Xicheng District
Tel: 010-64033225
Admission: $3