Of course you can travel on your own. But you'd better make your schedule before departure. It is very easy to tour by flight or train between those famous cities.
China is still one of the safer travel destinations in the world. Although petty crime, such as pick pocketing and purse snatching is on the increase, especially in major cities, serious crime against foreigners is rare. Also, the Chinese government guarantees the safety of the foreign travelers, even if relations between the visitor's home country and China are poor.
It doesn't mean that you shouldn't be as cautious in China as you would be anywhere else.