Trying to find a listing/database

Google is your friend.

BelahA wrote:

I am formulating a business plan that requires data on American cities.  I need a record of every state's cities

Every state's cities? 

There are 19,500+ incorporated and many thousand unincorporated cities in the US. I don't know any website where you can get that information aside from the US Census Bureau.

BelahA wrote:

and the corresponding post codes

There are 42,000 ZIP codes in the country.  (ZIP code is post code in the US.  ZIP stands for Zone Improvement Plan).  Many cities have multiple ZIP codes (LA, NYC, Chicago, Minneapolis, Houston, Philly, San Antonio, Phoenix, Dallas, etc. all have 40 ZIP codes each.)

There are some websites that give you a very small fraction of the ZIP codes online.  For a complete list, your only source is the US Postal Service who charges for the comprehensive data.