Hi! I'm Zayle. I found a $5 download for Civ III and thought it'd be nice to have it. So, here I am, with no Civ experience beyond the Civ Rev demo (and beyond that, no strategy game experience save a bit of AoE II), and no manual to go by for a very complex game.
Zayle, welcome to the Forums.
Some Random Hints
Play a game or two on Chieftain, the easiest level, to get a feel for how the controls work and where to find things in-game. Then consider moving up to a more difficult level.
Wonders are nice, but don't get fixated on them. You have to have the UN to win a diplomatic victory, but it is useless when you want to conquer the world. Only build the ones you need. Or let the AI build it and then capture that city.
In the same vein, don't build every city improvement you can. Build only what you need. In the very corrupt cities (ones that have a lot more red shields than blue shields) don't build any improvements at all. Instead, build workers, settlers and artillery pieces, since artillery units do not benefit from a barracks.
Except at the first of the game, use barracks to build veteran units. They attack and defend better than regular units and cost the same to produce.
Trade Reputation
Make trades carefully. Your trading reputation, how well you keep your word on trade deals in-game, is very important to the AI and a lot of things can break it and make it hardert to trade later. The hazard is when you trade a luxury or resource or promise to pay gold per turn (gpt) to the AI. For these deals, you must have a road network that connects your capital to their capital. If not roads, then a sea route. But, if an enemy AI or barbarian blocks that trade route over the In-Between Turn (IBT) then the trade deal is broken and the AI holds you responsible.
If the deal is just a tech swap (Iron Working for Alpahbet), it happens then and both of you benefit from the deal immediately. The same is true when you pay a lump sum of gold to buy a tech; you gain that knowledge then.
If you are in war and make peace and then reattack before the 20 turns of peace are done, this will also wreck your rep.
The thing to remember is that you are at risk when you are providing something to the AI for 20 turns, whether it is gold, a luxury or strategic resource. So just be careful.
Declaring War
When you declare war you can also damage your trading reputation. If you have any units inside the cultural borders of a civ and you declare war on that civ while those units are still inside their borders, you are a bad guy and ruin your trading rep. Thus, before you declare war, make sure all your units are outside your soon to be enemy's land, including your naval units.
This doesn't seem to affect the AI any.
Wandering AI Settler Pairs
Every so often you will spot an AI settler moving across the map protected by a Spear or Pike or Musket or Rifle. They will walk across your borders and stroll throughout your lands looking for a place to settle, one tile a time. You are expected to ignore this affront. But don't try the same on the AI. A settler or worker they will generally ignore, but add a military unit and they will kick you out.
Border Expansion
Your borders expand based on the culture points of your cities. In order for the border around a city to expand beyond the initial nine tiles, it must have a culture producing building in it. The cultural borders expand at 10, 100, 1000 and 10000 culture points for the city. The two cheapest culture producing improvements are Temples (60 shields) and Libraries (80 shields). Of the two, build Libraries. They are more expensive and will take longer to build. They also give more culture points per turn than temples (3 vs. 2) so the borders expand sooner. And they help you to learn things faster. Over the long run they do more than temples.
Succession Games
Read through some of these. As the players discuss various aspects of the game amongest themselves you will learn from real life examples of how the game works and what to expect from it. If you get the chance, join or start an SG. Since you are only playing 10 turns at a time, you can play at a higher level than normal, and in the time the higher level will seem normal.