If you plan on using the internet on rare occasions, maybe to check out your emails once in a while, then my suggestions would be to go for a free service. NETZERO and JUNO offer free internet access for 10 hours per month. This service is free of charge, but will come with an advertisement banner on the top of your webpage at all times.
If you feel the need for some more internet time, then you will fall in the category of an occasional/regular internet user. You may find comfort by just having a dial up connection because they are the most affordable starting from as low as $4 p/mo. to as high as $20 p/mo. (depending on the features that each plan has to offer). Dial-up accelerators are an attractive feature that allows web pages to download faster by compressing the information.
DSL connections cost around $20 to $40 per month, for users that feel the need for more speed in their internet connection. The advantage of using DSL is that it will not keep your phone line busy. COMCAST offers great coverage for DSL users and many features to upgrade if needed.
The final stage is for frequent/heavy internet users, usually those that will spend more than 6 hours a day online; the best alternative is either broadband or cable. This service will cost you more than any other type of connection mentioned before.