This is a great idea for a thread, thanks for starting it.
I've only played one full game on Civ III to the modern era of technology but the strategy which I favor for global conquest is as follows:
Explorer (key: Mapmaking Tech)-My first priority in the game, beyond discovering the initial environs where I'm placed is to make contact with every civilization I can. By doing so, not only do you know where everyone is located by you can also ascertain what technologies everyone holds. Due to unique civ traits all the civilizations start with different technology. If you can trade with technology with every civilization you know, you can end up with the sum total of all that technology. With this approach you will end up helping some rival civilizations with technology but ultimately you benefit the most by acquiring the best of the tech out there. Odds are, your rivals won't explore as aggressively as you limiting this strategy on their behalf. To get in contact with many civs you are going to need emphasis seafaring tech early on.
Homeland (key: Mobile Units)-Once you've determined what kind of land you have to work with, and who your neighbors are, its time to start thinking of yourself as a nation state. For a strong homeland, you have got to figure on needing about 10% (15 to 20 cities) of the world map. This is a must because otherwise you aren't going to have the military/industrial capacity to accomplish goals later in the game. Rarely will you start with enough open land in front of you for a large consolidated homeland. You have to identify territorial opportunities. Is there a nearby island that would make logistical sense for your homeland? How is the continent shaped that you play on? Could you add through military means all the land up to a certain junction like a bottle neck in the land mass? How strong are your surrounding neighbors? If you want to engage in an early military campaign you are going to want to have at least horseback riding with the idea of developing your technology toward Cavalry. Try to take as much territory as is logistically sensible early on. Avoid a far flung civilization which is going to be very tough to defend on low levels of technology.
Industrialization (key: Railroad Tech)-Once you have a large homeland its time to focus all your energies on developing what you have. Return your empire to a defensive posture and build from within. Focus developing your industrial capacity and technological capacity first while sprinkling in Temples and Coliseums as your population grows discontent. If your cities are industrialized up through your current technologies, build wonders in the top production cities. If one city exceeds the rest in production build all your wonders there. Don't waste resources upgrading an ever evolving military. Keep your opponents at bay using diplomacy, and continue to trade technology unless you are a clear front runner in all the tech areas. Railroad is a key technology for this period because it does 2 things. 1.It improves your industrial capacity. 2. It bolsters the defense of your homeland, so in the event of an invasion you can defend yourself.
Militarization (key: Automobile)-When you have the technology and industrialization needed to quickly produce a modern military its time to do so. A modern military by definition is one which contains transports, destroyers, armors and bombers. The more advanced military technologies I classify as post-modern....helpful but not critical. With modern sea, land and air power it greatly assists defense and expansion of your empire. To defend each city, you are going to want two quality land units at least. Then extra sea power like destroyers and battleships associated with your major cities. You can never build enough transports, as they are one of the first targets by the AI in a war. I'm a proponent of a strong air force to soften up the opponents ground units, and air power is the most effective way to control the seas. The idea is to build enough excess military units that you have enough not only to defend your homeland but also to mount an invasion.
Expansion (key: Armies)- Now that you have a formidable military, its time to think about expanding your empire and eventual global conquest. To be successful in global conquest it takes more than just a good military. It requires strong diplomacy and alliance management. I find that staying out of mutual protection pacts is the best way to stay out of conflicts you don't want to be involved in. If you have too many protection pacts and get dragged into too many wars, eventually your friends will make peace with your rivals and then declare war on you. Stay out of everything you can while focusing on your own military targets. Select the weaker, more convenient opponents first to fight. That way you'll have less resistance while waging war for quicker territory acquisition. Increasingly take on more challenging opponents as your military and territory power grows. Also the most dangerous opponent isn't necessarily the largest....its the one that is going to be the most difficult to invade or disrupt. In my game, I decided to leave a dangerous but remote civilization alone until the end because of the logistical challenge of trying to score a defeat, instead going after a civ even larger at first that was easier to invade. A key to having very successful and rapid military campaigns is having armies with the ability to produce them. Armies become your sole offensive unit, to go along with air power while regular land units fill in the gaps for mostly defensive purposes.
The statistics of my current (and only civIII game):
-88% of the global population
-150 cities
-Unit capacity of 900 (communism)
-18 armies
-Destroyed 12 civilizations
-1 remaining civ (Iroquois) with 12% of global population