In China the Internet is readily accessible. Internet cafes (网吧 wǎngbā) are abundant throughout China. Many of them are designed mainly for gaming though and are not useful places to do business. It is cheap (¥1 to ¥4 an hour) to use a computer, albeit one with Chinese software. Internet cafes are supposed to require users to show identification (passport), but this is generally not enforced. Traffic may be monitored.
It may be difficult to find an Internet-cafe with any service beyond simple access. If you need to use a printer or burn a CD, expect to search for the service, paying a fairly high price when and if you find it. The exception is tourist areas such as Yangshuo where these services are fairly readily available, though still at a price.
Most of the better hotels provide access from the rooms (often expensive) and/or provide a wireless service in public areas. Also, quite a few cafes provide free wireless Internet service — for example, Starbucks, Italy cafe, Feeling4Seasons Cafe in Chengdu, Padan cafe in Shanghai, etc. Some cafes, especially in tourist areas such as Yangshuo, even provide a machine for customer use.
A word of caution: public computers and the Internet lines they are connected to are not secure. Assume that anything you type in can be viewed by others. Do not send extremely sensitive data such as banking passwords from an Internet cafe.
If you are planning on connecting to the Internet with your own computer, be aware that many places (especially college campuses) require you to use Microsoft Internet Explorer and to install (censorship?) software on your system and/or accept certificates in order to use their services. For Mac OS or Linux users, look into using a browser that can fake its identity such as Opera.
If you have access to a corporate VPN outside of China, it will let you bypass the firewall systems. Another option is a service provided by Peace Fire. They provide unblocked access to all web sites without the need to install any software. You can subscribe to a mailing list and always stay up to date about it. Tor software may also be of great help.
Bypassing the firewall using a VPN is a safe activity. However posting information that the government considers subversion is taken seriously, and Internet companies often help them. Both Google and MSN have agreed to censor in order to get Chinese licenses. Yahoo! went further; in one recent case they turned in a Chinese user who got ten years in prison! While travelers are generally not at risk, it would be sensible to be cautious. In particular, beware of getting Chinese friends into trouble.