Officially, the most important issue for the Chinese government is corruption. Each year, the government prosecutes more officials for corruption. It was 2,687 last year. But it wasn't enough, as the number of corrupt officials is over a hundred times that. All countries have corruption, but if there is too much of it, there is inefficient government and more unrest. Most of the 70,000 officially noted public demonstrations and riots each year, are caused by the corruption of local officials. Senior officials know, and publically recognize that the widespread corruption is bad for the economy. Privately, they see corruption as the major threat to continued Communist Party control of the government. At the same time, a majority of the lower level Communist Party officials prosper only because they exploit the people they rule (usually at the town, county or province level). These officials steal taxes, take bribes and falsify land ownership records. They are very unpopular, and the national government can't prosecute all of them. There are simply too many. Urging them to reform themselves has not worked either. There's also a lot of corruption in the military (always has been, even before the communists took over), but this is seen as less of a threat to the government. In fact, it's more important to keep the military happy, in case there is a major uprising.
The recent harassment at sea incident off Hainan is apparently China trying to gauge the resolve of a newly elected U.S. government. The last time there was an incident like this was in early 2001 (when Chinese navy jet fighters harassed a U.S. Navy electronic monitoring aircraft off the Chinese coast.) If the U.S. government folds, the Chinese will keep pushing. But, as happened in 2001, the Americans pushed back, and that was the end of that. The U.S. responded to the recent Hainan incident by sending a destroyer to escort the U.S. survey ship, and warning China that force would be met with force in international waters (the "economic zone" is considered international waters as far as safe passage and using sensors is concerned, although China interprets international law differently.) China agreed, however, that further confrontations would be avoided.
The Taiwanese government is increasing defense spending, including the development of more locally made weapons (like cruise missiles to take out targets on the mainland.) Chinese pressure on the U.S. caused Taiwan's recent request to buy 60 F-16 fighters, to be turned down. Taiwan wants to increase its military capability because its arrangement with the United States requires that Taiwan be strong enough to hold off a Chinese attack long enough for American forces to arrive. This means keeping control of air bases on the island for up to a week. China is apparently building up its land, air and naval forces to the point where a surprise attack could conquer Taiwan in a few days, if the defending Taiwanese were not ready.
China is trying, without much success, to convince North Korean officials to back off its military spending, and devote more attention to economic reforms. But the North Korean Communist Party members are obsessed with how quickly (and sometimes fatally) communist party officials lost power in Eastern Europe in 1989. The communist party rule in North Korea has been far more brutal than it ever was in China. The North Koreans are more of afraid of, and less trusting in, their own people. China does not want a rebellion in North Korea, because millions of refugees could end up in northern China, and China would have to spend billions of dollars taking care of them, and dealing with the unhappy Chinese population in the area.
The Chinese military graduated its first class of Public Affairs Officers. Western forces have had such media professionals for decades, but in China, propaganda/media specialists from the Chinese Communist Party have long handled the media needs of the military. But now the military will increasingly handle its PR itself.
March 19, 2009: In an unusual incident, a soldier on guard duty at a Central China army base was shot and killed, and his assault rifle stolen. Another soldier in the area was shot and wounded. The army has to deal with thieves and criminal gangs stealing from bases, but it's rare for there to be a direct attack on soldiers like this. The area where the attack took place has many ethnic Tibetans, and some of these are suspected of carrying out the attack.
March 9, 2009: Chinese police and soldiers are increasingly visible all over Tibet, ready to arrest anyone daring to do anything to recognize the 50th anniversary of a major uprising against Chinese rule in Tibet. The police are particularly intent on catching foreign journalists in Tibet illegally.
March 8, 2009: About 140 kilometers off Hainan island, five unarmed Chinese navy ships tried to chase away an unarmed U.S. Navy survey ship (that had deployed sonar equipment). China considers surveying or monitoring within its 360 kilometers "economic zone" to be illegal. This is the Chinese interpretation of the international law that gives it ownership of raw materials on the ocean bottom, and fish in the sea, within the economic zone. But in China, surveying (for maps) can only be done by the government, and non-government surveyors (who are increasingly active) are sought out and prosecuted.