The Presidential Palace. Historically the site of a number of important government officials, it is also a beautiful garden, an oasis from the noisy, bustling city. About ¥40 a person.
Purple Mountain. Although the base is crowded, for ¥60 a person, there is a chair lift to the top which is scenic. Again, a nice getaway from the city.
Suzhou has many water-towns. These are cities interlaced by rivers and streams. The main means of tranportation was once by water. Zhouzhuang is the most famous of the water townships.