Usually the safest way to do this is to use Winzip. Unzip the file you download to the default C:\Unzipped folder. There will usually be a readme of some kind, especially with the bigger files - make sure to carefully read it. Saved games usually do not have a readme included, keep that in mind.
I do not currently have the game installed, so I can not point you to the exact location of Civ III - C:\Program Files\something or another\CivIII. Within this main Civ III folder is a whole bunch of other folders. One of them will be your saved games folder. Open that.
You can now cut and paste the files from the unzipped folder to their proper location.
As has been mentioned, if you are going to install something big - like a mod or a graphics pack, make sure to backup the original folder. To do this, right click on the mentioned folder and select 'copy'. Find a different location, let us say that C:\unzipped folder, open that - right click on an empty section of the folder - select 'paste'. The copy should now be safe and sound.
If you install the mod, or graphics and decide you do not like them, you can easily put the older one back.
Hope some of this helped.
