Frankly I think th biggest lesson learned by the US military in Iraq is that the historical trend indicating that insurgencies ultimately prevail has not yet been broken, which is something China might want to consider when contemplating an invasion of Taiwan.
I wold have to disagree with this. Historically, most insurgencies fail. The U.S. has won most of the guerilla wars it has fought in in modern times. The big exception is Vietnam. The US defeated insurgents in the Phillipines during the Phillipine Insurrection and the Moro Rebellion, they defeated Nazi insurgents following the German surrender, they defeated insurgents in two campaigns in Haiti and two in Nicaragua, and one in the Dominican Republic. They also fought irregulars in Mexico. The US has also prevented insurgencies from becoming large and from violence from spreading four times in Cuba before WWII. The US has also provided advisors to other countries where the counterinsurgency operations there were successful. The exceptions are Vietnam, and the fight against the Bolsheviks in Russia.
The biggest factor to American defeat in Vietnam was not the fact that our main opponents were guerillas waging that style of warfare. It was that the strategy and tactics used were more appropriate to conventional conflicts than to small wars, and in fact were entirely inappropriate to fighting guerillas. This was in no small part due to Westmoreland's background, as well as his ambitions. Proper strategies were used on a limited scale in certain parts of Vietnam, and they were highly successful (CAP comes to mind). They were limited, though, by Westmoreland. The US ignored experienced men who had fought insurgents as well as the USMC (which has a doctrine for fighting such conflicts borne out of the lessons of fighting and winning multiple such conflicts), as wellas other counterinsurgency experts such as Sir Robert Thompson. Our strategy, which was a conventional one, wore out our forces which in turn also wore down the people back home (although this took much longer than during the Iraq conflict, as the American people now seem to have an irrational intolerance for any casualties), and this ultimately caused a political defeat which resulted in withdrawal, and the goals of the conflict were not ultimately achieved. Unfortunately, the wrong lessons were drawn from this conflict. One camp believes that the lessons are that we should have gone in full force, unrestricted, and destroyed everything (this is actually very counterproductive), and the other drew the conclusion that such conflicts are unwinable and best avoided. Both camps are wrong, and history proves that using proper strategies and tactics, victory is very possible, although it is rarely instant.
In the case of Iraq, improper strategies and tactics were used from the start, although not to the dgeree in Vietnam (thankfully), and this is why the insurgency has been so hard to fight. General Petraeus, who wrote the Army's first counterinsurgency manual (the USMC has had one since 1935, with a new edition recentloy published), has been rapidly implementing proper strategies and tactics (although not to the fullest extent), which will produce results, although it will not be instant in most parts of Iraq. These strategies and tactics should have been used from the beginning, but the Army, lacking counterinsurgency experience using proper strategies and tactics, is ill-suited for the task, and leaders were not trained to fight in such a manner, and are trained often to look down upon such a form of warfare. If the politicians can be prevented from causing defeat in Iraq, the ultimate lesson will be that they can be won (guerilla wars) by armies, provided they use the proper strategies and tactics.
In the case of the PLA, they have shown no inhibitions to being downright brutal and repressive, and this is the only other way they can win, although if not done harshly enough, it can backfire. That is something they should consider if they try to take Taiwan. They should already have a set doctrine (very brutal, or carrot and stick) before they go in, one that their men have been at least rudimently trained in and the officers extensively. Tere should also be an increased emphasis on small-unit tactics. I must admit that in this situation I'm biased, since I feel that the ROC government should be the ones in Peking, not the PRC, and I would hope that the PLA would be defeated in such a conflict, although this does not change what the PLA should do to win. Taking Taiwan would be a very tricky situation, especially if there is US assistance, even if it is only indirect (to include things like providing arms, advisers, equipment, etc. to insurgents). It is an interesting scenario to consider, though.