One of the easiest ways to find surrounding zip codes to a specific zip code is to visit the Zip Code Finder and Boundary Map at http://maps.huge.info/zip.htm. This site is an excellent resource to find out which zip codes surround another zip code of your own specification.
There are two ways to search for a specific zip code and its surrounding zip codes on this website. The first way is the enter the zip code you wish to find. The other way is by entering a specific address. The latter is especially helpful if you have an address in mind but do not know its specific zip code.
Once you enter a specific zip code, the map will zoom in and show you, in color coding, all the zip codes that surround it. It is even overlaid on top of a normal map, which helps you determine the street boundaries for each specific zip code. You can also click on the individual zip codes, and as you do they will pop up beside the map on the right with some basic information: the zip code's corresponding city, corresponding state and estimated number of households, according to 2008 estimates. While the household numbers may be a little out of date, they can still give a general idea of the population of each zip code.