Finding Telephone Numbers
A good way to find telephone numbers is to use a free "411" service:1-800-FRE-E411 (1-800-373-3411) This corresponds with the online service called "Free 411" and although it is free, you may have to listen to an advertisement to use it! (The online directory also works with a free telephone service called "Skype.")
Traditional directory assistance services that use an operator generally charge a fee, as does the standard telephone "411" service (these two links bring you to Wikipedia entries; while Wikipedia is useful for general information, it is an easily-edited resource and information should always be backed up with reliable sources). The 411 online component is a free directory and is provided through the White Pages.
With any service you choose, knowing the area code is helpful (as well as the city and state of the number you are seeking). If you don't know the area code, try using the 411 AreaCode and Zip Code finders.
There are free online alternatives, but none of them are as up-to-date as directory assistance. If you need the phone number of a business in the United States, try a search on SuperPages.com or on the Yellow Pages.
For other phone number sources good places to start are the Phone Numbers section of Yahoo and the IPL's listings of Telephone & Address Resources. For directories outside of the United States, try Telephone Directories on the Web.
Reverse Telephone Directories
You may also want to look at the Internet Public Library's FAQ on Finding People.