Internet access is everywhere, so this is pretty easy to take care of. Membership at the local library will give you some time of free Internet access. If you have the money to patronize the business, many coffee shops and delis will have one or two computers for their customers to use.
If you can afford it, a laptop computer with a WiFi card will guarantee you in most major cities. Hotel lobbies, many restaurants, and increasingly today some areas just out on the street have WiFi Internet access in them. Purchasing a high gain directional wireless antenna can make it easier to acquire signals. There is a recent (as of 20060528) case however of someone being arrested for connecting through someones wireless access point without permission. I prefer to ask people first, but this complicates parking. Some people have recommended that owners of WAPs add "please use" or something to the ESSID string that identifies the network in order to give explicit permission.
The ultimate solution is a laptop with a satellite modem, enabling you to connect from anywhere in the world. However, this can cost some money, particularly with an ISP to deal with.
There are also some cellular Internet options that are increasingly cheap and fast.