There's certainly plenty to see in Beijing; I've spent a total of about 2 months there, and there are still a lot of things I haven't seen that I want to. Obviously go to the Forbidden City (Gugong), Temple of Heaven (Tiantan), the Summer Palace (Yiheyuan), the sites surrounding Tiananmen square, the Great Wall (the Mutianyu section in preference to the more popular Badaling section), the Ming tombs etc... these tourist sites are great, but you'll get sick of the crowds fairly quickly.
I'd advise going to the old Summer Palace (Yuanmingyuan) - not that much left to see, but it's a very pleasant, laid-back park, and you'll get a good view of typical Chinese park life, without the overcrowded feel. The Tibetan Lama Temple (Yinghegong?) is the largest Tibetan temple outside the Tibet/Qinghai/Gansu region. Beihai park attracts the crowds, but the many other smaller parks are great (try Taoranting, or the much-improving zoo). The moat has been reconstructed in the last two or three years, and makes a pleasant (if somewhat concreted) alternative to the open sewer that it was a few years ago. Go for a walk around the little lanes (Hutong) that constitute the more traditional side of Beijing; they are rapidly being destroyed to make way for new appartment blocks, but there are still some surviving. There are other sites outside town that are quite easy to get to - the site of Peking Man at Zhoukoudian, the Fragrant Hills (Xiangshan), the reservoir park in Heping county to the northeast etc...
Trips of a few days that can be recommended: try going to the beach at Beidaihe, and the nearby beginning of the Great Wall at Shanhaiguan. The palace at Chengde is a popular weekend trip. Alternatively go to the edge of Shanxi province, and visit the Buddhist sacred mountain of Wutaishan, with its multitude of temples and monasteries (my personal favourite). The ancient capital of the Wei dynasty at Datong (in Shanxi) is only 8 hrs on the train, and has the spectacular Yungang grottoes.
Trips of a week or more might involve seeing the sights of Xian, Shanghai etc... but don't be too ambitious, as travel in China is slow and buying tickets is even slower. Why rush, anyway?
Staying in Beijing: the big hotels are fine if you are made of money, but if you consider $10 to be a lot (like me), then there are an increasing number of small places springing up in the last few years to cater to less-well-heeled foreigners. There used to be just one, the Qiaoyuan Hotel near Yongdingmen station (now Beijing South Station), but the service there leaves a little to be desired. But you should be able to get a reasonable room for less than 80 yuan (about 2 weeks salary for a Beijingren), and a dorm bed for considerably less.
Food: you won't need Imodium these days (though you might want to take some if you're really paranoid) - stick to the local yoghurt instead, which comes in little porcelain bottles with a waxed paper cover, and will protect your stomach from the bugs as well as refreshing you. Beijing isn't quite up to the standards of Guangzhou or Chengdu for food, but it beats anywhere in the America for variety of flavours, prices and types of food. I'd recommend you try the northern staples of jiaozi (stuffed dumplings) and baozi (stuffed buns) as well as the better known/more pretentious stuff like Peking Duck (Beijing Kaoya), great though it is. The street food is variable in quality, though if it's cooked in front of you it's fine; the shish kebabs (Yangrouchuar) make great snacks.
A few notes of caution: like most modern Chinese cities, much of the place appears dusty, drab and grey; it's worse in winter. Bear in mind that the place would basically be semi-desert without the city (the only capital in the world not built on a river?). Just avoid spending too long in the streets; the parks and sites are usually much pleasanter, as all incorporate trees and water. The city is big, therefore taking the bus or getting a bike is the best way around. Bike hire is fine for a few days, though if you are there for a month you might try buying a Flying pigeon of Phoenix bike, and reselling it at the end (bike demand still exceeds supply in China). The repair men on the street side are generally fairly efficient, though if you're charged more than a few dollars, he's really trying his luck. If you're not familiar with Chinese ways, just be tolerant, patient and retain your sense of humour. The Chinese approach to a problem is usually to wait and see what happens; surprisingly this approach does work, even in the most unlikely situations.
Be careful when taking advice about the place before you go - the place is changing extremely rapidly. I first spent a month there in 1987; when I went back in 1990/91 it was almost a different place. In 1993 the place was full of yellow taxivans; when I tried telling other foreigners that there weren't any at all in 1992, they just wouldn't believe me.
Have a good time! I wish I was going back too!