I don't know how you installed your current Zen Cart, but I can tell you how to proceed with a manual install (the best way). I used Fantastico once a loooong time ago to install Zen Cart, so I don't remember how it works.
If you're going to get a new domain name for the new store, do that first and either add it to your current hosting (as an add-on domain or sub-domain, if your current hosting allows) or get separate hosting for it.
Create a database for the new store from your webmaster panel. Make note of the database name, user and password. You'll need them later in the installation process.
Install Zen Cart for the new store by FTPing it to the new location and running zc_install which will ask you for the database info.
After that, you could use phpMyAdmin to back up the database from the current store and restore it to the new one. That'll copy all customer and product info to the new store, but it'll save you setting up the products all over again if you'll be selling all the same merchandise.
If you elect to share the database, change made through the admin will effect both sites. For example, you turn on/off a sidebox. It'll be on/off on both sites. If you add a product, the product will appear on both sites.
Mary Ellen
I came; I saw; I Zenned
Taking over the world... one website at a time
Make sure brain is engaged before putting mouth in gear... or fingers to keyboard.
Holzheimer
Fan Odyssey