Beijing

I think the closer to Chang'an Dajie and Qianmen...the more expensive it'll be, since you'll be right by the tourist attractions (Forbidden City, etc)-- if you can get decent priced hotels there, then I think that's fine.  But, in general, you can get anywhere by taxi, so it's not really a huge deal to find one in a super-prime location.  Here is my breakdown:

    * Dongcheng District (东城区)-- I used to live here; good location-- on the east side, but not so far as chaoyang (East Gate Plaza is near a couple of subway stops, which might be nice for you)
    * Xicheng District (西城区)-- Older part of town, the Drum Tower (gulou or jiu gulou) are good places to stay)
    * Chongwen District (崇文区)--- No, unless you're staying between the Temple of Heaven and Tianamen
    * Xuanwu District (宣武区)--- NO
    * Chaoyang District (朝阳区)--good for foreigners... where the CBD is and World Trade Center, basically the business areas
    * Haidian District (海淀区) ---NO, this is far from the center....you'll go this way to see the old summer palace, but don't stay so far out.
    * Fengtai District (丰台区)---NO
    * Shijingshan District (石景山区)--- I don't even know where this is! So, I'll say no.

Whenever I traveled around China, I stayed at the hotel chain "Home Inn" (如家).  It's super no-frills (like Motel 6, it's a "2 star" hotel).  Tiny rooms.  But, whenever I stayed, I thought it was clean, and fine for my purposes.  It's cheap, and there are a few in Beijing, I think.  Anyway, in general, I would say, stay north of Qianmen/Forbidden City (rather than south), and north or northeast of the city, rather than in the northwest.  I just think there are more things to do there...

I also used to book through "sinohotel.com"--it was just easy to do, so it might be useful to poke around there?

In general, though, everywhere you go, there will be lots of food choices available.

The Peking Duck place that I used to go to is called:

Xiang Man Lou
香满楼
Tel: +86 10 6460 6711
Address: West of Yuyang Hotel, Chaoyang District
朝阳区渔阳饭店西侧

Other: Where to go? If you can travel to the west city near the CCTV Tower (China's tallest building) you will find JiuHuaShan to be the best choice in the city. It is located in a hotel complex on Baiwanzhuang Road heading west before it reaches the Third Ring Road. However, if you are on the East side of the city, the best option is: Beijing Duck Chuan Lu Feng Wei on Xinyuanxili Road (east) across from the YuYang Hotel. Ask your hotel concierge to show you on the map these two top locations (non-tourist places where all the expats like to go). The above mentioned Ritan Park Restaurant is also a good place for a duck for lunch or dinner - 58 RMB.

Unfortunately, the best dumpling place is still Din Tai Feng!  Since you probably don't need to go there, I would check out Dongbeiren since they have  lots of tasty colorful dumplings:
Dongbeiren
The three-story restaurant's raucous, colorful atmosphere captures the swaggering spirit of Dongbei. Along with the squirrel fish and the gamey venison, its hard-drinking, loud-talking clientele goes with the territory. Daily 11am-10pm. A1 Xinzhong Jie, Dongzhimenwai, Dongcheng District. (6415 2588/2213) 东北人, 东城区东直门外新中街甲1号
(http://beijingexpert.com/beijing-restaurants/chinese-food-in-beijing/)

Lamb Hot Pot 涮羊肉 Shuang Yang Rou
Recommended restaurant: Dong Lai Shun 東來順王府井大街新東安商場5樓
Address: 5th floor, New Dong'an Mansion, Wang Fu Jing Street
Phone: 010-65280932 / 010-65280501
Directions: Dongcheng District

Sites

source: travelforkids.com
The Great Wall 

about a 1.5 hour bus ride from the city. It is possible to rent a taxi for a day to take you to these sites. Renting a taxi should cost ¥400-450. For this price the driver takes you wherever you want, and will wait for your return.

The most popular place to visit the Great Wall is Badaling, but Mutianyu is your best bet with kids. Mutianyu is less crowded, and it has a cable car that takes you to and from the Wall – the ride itself is fun and is helpful if you have little kids (save their energy for running up and down the Great Wall, not walking to it). Badaling also has a cable car, but if you opt to walk to the Wall, it can take 45 min. to 1 ½ hours.      To visit the Great Wall from Beijing, the easiest way is to arrange at your hotel for a car with driver.

Badaling (八达岭 Bādálǐng): Give yourself about 2 hours to climb up the wall if you're a slow climber, an hour or less if you're speedier. The least expensive way of getting there is to take the Bus #919直快 from Déshèngmén (德胜门), just east of the Jishuitan Line 2 Subway stop. The journey lasts about one hour, and costs around ¥10. There are also official tourist buses that leave from Dongzhimen, Qianmen, and Beijing Railway Station. Tour companies charge ¥150 and usually stop at a jade factory, and one or two other stops. They do provide an English speaking guide and lunch which can be advantage. Another, not especially convenient, option is the train: the 8:06AM Shāchéng (沙城) train leaves Beijing North Railway Station daily, and gets to Badaling at 10:49AM. Return trip leaves at 14:34. The final option would be to hire a taxi for the day. It's common to have a 4-hour package from a taxi running somewhere around 600-800RMB (the driver will pay the tolls). Be sure to get the driver's cell phone number in case it's needed. Best bet is to hire a driver using a trusted tour guide company, commonly found in hotels. Be sure to collect a receipt of your purchase as well. Near the entrance of the wall it feels like you've entered a ski town, with some gift shops and coffee to warm up to during the cold winter months. Tickets to climb the wall are around 40RMB, discounts for young kids and the disabled.

Tiananmen Square

            Tiananmen Gate - entrance into Forbidden City

           Mao Zedong Memorial Hall - long line moves fast, but not for antsy kids - can't bring in cameras and MUST be quiet

            Fly kites

           Wax Museum (China National Museum)

Imperial Palace = Palace Museum -  Forbidden City

Get there when the gates open (around 8.30am) if you want to walk through the vast and spectacular courtyards in relative peace. Current entry fee is ¥45. You can rent an audio guided tour which explains certain temples and their uses for ¥40 with a ¥100 deposit (more than a dozen languages available). The signs posted around and on buildings inside are fairly short and are written in both Chinese and English. You could reach the Forbidden City via subway line 1 in the station of Tian An Men West or Tian An Men East,the bus station in the same name for line 1,4,52 and 728 is also convenient for catch in the Chang An Street. Alternatively, the bus station of "Gugong" for line 101,109 and 810 is sitting in the north gate of Forbidden City.

         lion statues, hall of supreme harmony, dragon pavement, galleries, etc

        tip: go early in day to avoid crowds

        jingshan park (out north side) to look from hill over forbidden city

 Beihai Park – Beihai Park is the perfect place to go after you've seen the Forbidden City, and it's just a great place for kids. Once an imperial pleasure garden, Beihai Park is peaceful and relaxing, cool and shady, you can escape the crowds, there are benches for picnics, and places to get ice cream. More than half the park is the lake, so rent pedal boats (kids can expend some energy) or electric boats (if you're feeling lazy).
                                       

Walk across the white marble bridge to Jade Island, climb up to see the White Dagoba, which has Buddhist relics buried inside, and for views of the Forbidden City in the distance. Before the White Dagoba was built, this hill was the site of Kublai Khan's Palace of the Moon in the 13th century. Marco Polo might have sat on this very spot. From the northern edge of the islet, you can pick up a ferry across the lake.
                                      

On the northwest corner of the lake, you'll find a children's playground, the Nine-Dragon Wall, a stunning glaze tile wall with nine lively dragons frolicking in a wavy sea, and five waterside pavilions, connected by little bridges.

         
South of Tiananmen Square
           
Northwest

Purple Bamboo Park (Zizhuyuan Park) – This one of Beijing's prettiest parks, three lakes and two islands, connected by lovely bridges. The lakes are densely filled with flowering lotus plants, and park is planted with bamboo and other flowering plants. Popular with local Beijing families, there's also a children's playground and small amusement park.
           
Summer Palace
(颐和园 Yíhé yuán) extensive gardens and the ruins of palaces constructed by the Qing emperors. Most visitors stay in the front hill area, but if you prefer quiet places, the west bank and back hill areas are good choices. There are some quiet and secret ruins, caves, and other fun stuff in the back hill area. You could easily take bus No. 690 or 808 from Tian An Men Square to the terminal station, the Summer Palace. The Bus 826 would bring you from Temple of Heaven to Summer Palace.           

Kid Centric

source: gochina.com 

Fundazzle

• Description: Indoor playground, ball pit, playgym, etc.
• Address: 1)Gongti Nan Road; 2) Worker's Stadium
• Opening hours: M-F 9am-5:30pm, S-S 9am-7pm
• Admission: 30rmb (US$3.75) per kid
• Comments: Take it from me (who has spent hours there with her toddler), Fundazzle is the perfect place for kids on a rainy day. Take them there, exhaust them for several hours, then stick them in the stroller and off you go to the Forbidden City.

Beihai Park

• Description: Large historical park northeast of the ForbiddenCity.
• Address: 1 Wenjinjie
• Opening hours: daily 6:30am-8:30pm
• Admission: 10rmb (US$1.25)
• Comments: Beihai Park is a fantastic spot to amble around enjoying a break from the city. Climb up to the white tower, a Beijing landmark, or have fun on the paddle boats.

• • Description: "Wu Se Tu Gongyifang" in Chinese, an interactive center for arts and crafts.
  • Address: Dog Cheng Shaonian Gong, Dong Zhi Men Nan Lu 10, East side of East Second Ring Road, green-roofed building north of metro stop, Dongsi Shi Tiao
  • Opening hours: daily 9am-6pm
  • Admission: 50rmb (US$6) for 1 hour, 80rmb (US$10) for 2 hours
  • Getting there: metro stop, Dongsi Shi Tiao (213, Exit B)
  • Comments: Children can try their hands at tie-dyeing, embroidering and painting as well as pottery (though for that you'll need to book in advance 010-6415-3839, have your hotel concierge help).

Looking for toys in Beijing? I recommend The Children's Department Store on Wanfujing right near the Beijing Hotel. It is one block north on the east side of the street near MacDonalds.

Apartment

Located along the East 2nd Ring Road, right besides the Swissotel, it provides easy access to the Sanlitun Embassy Area.

19 Dongzhong Jie 

Situated between Dongsishitiao Station and Dongzhimen Station, East Gate Plaza is right beside the East Second Ring Road and directly north of Poly Plaza.

East Gate Plaza locates in Dongzhimen hub of Dongerhuan Road, between Dongzhimen flyover and East Sishitiao flyover. It is surrounded by Hong Kong and Macao Center, Baoli Mansion and Oriental Ginza with the Worker's Gymnasium, Second Embassy and Sanlitun Bar in the east.
East Gate Plaza is quite near to Sishitiao subway station of the east line, subway line No.2,
Dongzhimen subway station and the crossing of Dongerhuan road and Pingan Avenue, Dongzhi stereoscopic hub of communication is under construction. There is planned subway which runs to airport.

Airport

Many people use taxicabs to reach town from the airport. Try to get the Chinese name in characters of your hotel so that you can let your taxi driver read where you want to go. A taxi from the airport should cost ¥70-120. You will have to pay the fee shown on the meter (make sure the driver uses it) plus ¥10 toll for the airport expressway. Traffic jams are common.

The cheapest way to get to the city centre is to take the airport shuttle [17] (+86 10 64594375/64594376) . There are several lines running to different locations throughout Beijing. Buses for each route leave every 10-30 minutes, and tickets cost ¥16 for a one-way trip. Follow signs for the buses (机场巴士 Jīchǎng Bāshì) to find the bus-stops: