A week near Hong Kong
Of course you could easily spend a week, or several, in Hong Kong itself.
Also, there are several nearby areas of China that are well worth visiting:
There are direct flights and/or overnight buses to all of those.
Here I assume you want to stay within a few hour's bus or train travel of Hong Kong.
The region near Hong Kong is called the Pearl River Delta. Politically, it is three areas, parts of Guangdong province and two former colonies that are now Special Administrative Regions of China, Hong Kong and Macau.
Economically, it is a powerhouse. Guangdong does a third of China's exporting and is China's richest province. While Western countries have GDP growth of 1% to 3% annually, China has had 8% or better every year for a couple of decades. For the Delta area, annual growth has been around 15% for most of those years. The Delta has been the fastest growing region of the fastest growing province of the fastest growing country on Earth. As of early 2009, however, the Delta is also the area of China hit hardest by the world economic crash. Quite a few factories have closed due to a slowdown in export markets.
A high-speed rail system (300 km an hour top speed) is being built to link all of the Delta cities together. Parts of it are already in service.
Citizens of most countries can get Macau and Hong Kong visas at the airport, but will need to get a Chinese tourist visa in advance. You can get these from a Chinese embassy or consulate in your own country. For most passports, visas can also be obtained in Hong Kong or Macau via travel agents or the government office. The Macau office has shorter lines.
There are special visas which allow you into Shenzhen or Zhuhai but not out into the rest of China. They would not be useful for my suggested route, but might be for related trips. They may be obtainable at the border, but last I heard were not available for US or some other passports. Check the Shenzhen entry for details.
Guangzhou is the provincial capital a few hours up the Pearl River. It was known as Canton in the tea-clipper era. Guangzhou is thousands of years old and has always been the most important city of Southern China.
Macau is a former Portuguese colony on the West side of the river mouth. It was the first European enclave in East Asia, and a major trading port from its founding in the 1500s. Today its main attractions are casinos, European food and wine, and old colonial buildings. In 1999 it became a Special Administrative Region of China.
Hong Kong is a former British colony on the east side of the river mouth. It was founded in 1841, Britain's prize after one of the Opium Wars, and soon eclipsed Macau as the center of the China trade. In 1997 it became a Special Administrative Region of China, retaining much of its laissez-faire capitalist energy under the slogan "one country, two systems".
Shenzhen is on the mainland next to Hong Kong. It was basically a fishing village until the 1970s. Then it was made a Special Economic Zone and developed incredibly rapidly. Today it is a boom town of at least six million.
For an explanation of the terms "Special Administrative Region" (SAR) and "Special Economic Zone" (SEZ), see List of Chinese provinces and regions.
The traveler on a really tight budget should get out of this area altogether and seek the real bargains in China's hinterland.
The cost-is-no-object traveler can stay in Hong Kong, or go to up-market places in other cities, and have a wonderful time either way.
As a general rule, both Shenzhen and Macau are cheaper than Hong Kong.
Zhuhai is cheaper than any of the above. Consider doing most of your shopping there.
There are a lot of pickpockets throughout this area. See the Wikitravel article pickpockets for advice on precautions.
See individual city listings for other risks.