Hmm.. the only method i see is to use what the game gives you, but even that get people frusterated. However, have you looked at the Manual that Civ III came with?
Page 179 talks about what ports should be opened when it comes to:

-Hosting Internet Games
-Hosting LAN Games
-Joining LAN Games
-Joining Internet Games
-Joining via Direct IP

I'm not the smartest when it comes to computers, but i hope it brought you one step closer to the right path

I bought CIV III through Steam. Is there a digital manual? Besides most of the "original" methods replied on GameSpy hosted servers, which i don't think exsist anymore. It seems I am able to join some peoples' games, but most people I know are not able to host for each other.

Yes, there is a digital manual. when you select a game in the "Library" tab, right click the selected game instead of left click. One of the options should be to view the manual.

The manual is a PDF file and each 'digital page' contains two pages of the manual.
So if you wanted to look at page 100 on the manual, go to page 50 on Adobe or whatever you use to read pdf's