Some general tips regardless of what victory conditions you are playing for (I tend to go for peaceful victories, but sometimes have "accidentally" achieved a domination victory because the other civs didn't like the fact that I was getting close to the one of my goals (Cultural or Space Race) or had some resource they didn't and they decided that it would be a good idea to try and take it by force).
#1: Expansion is the best policy... Granted, I've started on some weird maps where all the AI civs' start points were about 5 tiles away from mine, and of course, the game always puts YOU in the middle of the AIs, so they had plenty of space to expand OUT, but they all decided to expand IN, to put the pressure on me... That just resulted in some early wars of one or more said AIs getting their butts kicked... The reason is that in such a situation, I forewent almost any kind of cultural production, and just started cranking out troops.
#2: Even in builder mode (trying to get a peaceful victory), I always ALWAYS never build two Buildings/wonders/small wonders in a row... My build order in a typical city is building, unit, building, unit and sometimes Unit, unit, building, unit unit, building... Later in the game, when Artillery actually starts becoming effective, I try to go: Unit, Artillery, Building, since Artillery can be used to greatly multiply the effectiveness of what troops you have, which brings me to:
#3: Artillery, Artillery, Artillery! Starting with Trebs, begin building up your Artillery stockpile. While they might seem pointless at first glance (after all, they cost almost as much as a standard unit and can't actually KILL any enemy units, right?), my biggest mistake when I started out was not building enough (or any) artillery.... The thing about artillery is that while they can't KILL an enemy unit (except in a few special cases, such as the Korean Hwa'cha), Artillery CAN reduce enemy units to 1HP, and if you have enough artillery to reduce ALL the defenders to 1 HP each, then you've basically already won the battle... The way Civ III's engine works, even if you have WARRIORS and they have INFANTRY, if you've redlined their infantry with your artillery, your warriors will still have a decent chance to defeat the enemy infantry... After all, they have to wipe all 4 HP off your warrior, but your Warrior only has to get lucky once.
For a good illustration of how effective this tactic can be, and a good way to come back from a VERY disadvantageous position, check out my Age of Imperialism scenario story where I took the Boer Republics and managed to take over almost half the world (thus far... the game isn't finished yet)... In the AoI scenario, the Boer Republics have some of the weaker unit lines (my units are weaker than some of the European factions' COLONIAL troops), limited resources in their starting area (no coal, so no railroads), and they are completely surrounded by colonies of several Colonial Powers (Britain, Germany, and Portugal), and are behind the Major Powers in tech (not to mention that your starting economy sucks, so you'll fall even further behind soon after the game starts).
Which basically means:
#4: Even if you DO fall behind the rest of the pack, don't give up! There are ways to catch up... One is to try researching a branch of the tech tree that nobody else is so that you can trade up, or you can use Ye Olde Intelligence Service (or Diplomatic Service) to STEAL tech. You can also use the "pointy stick" approach... E.g. build up your army (with plenty of artillery support) and go after the most advanced opponent... If you knock them down a few notches (or even better, completely out of the game), you'll no doubt find yourself catching back up in a hurry (I was so far behind Britain in the AoI game that they were almost a full era ahead of me by the time I went into all-out war with them... By the time I kicked them out of the British Isles, I'd caught up to their tech level, and by the time I wiped out their last enclave in New Zealand, I'd passed them in tech, and was on par with the surviving "Major Powers").
Finally, don't be afraid to try doing things differently... As I said before, I usually play peacefully, but Age of Imperialism is such that anyone who tries that route (except Britain or France with their gazillion colonies and the ability to just kick back and watch the Raw Materials, a.k.a. flag units, roll in and the victory points rack up until they get the requisite 4000) is toast, especially the minor factions (Latin American nations, Persia, Thailand, Boer Republics, etc). My immediate goal was to build up my economy and military simultaneously, and immediately go after either Mozambique, Angola or German SW Africa (Going after Britain right off the bat would have been suicide), and a war that broke out between Germany and Russia when I was almost ready to strike provided the opening I needed...
Once I "liberated" German SW Africa, I used those two newly acquired cities to ramp up my economic and military build-up, which, in turn, allowed me to capitalize on a war between Britain and Russia a little while later, at which point I "liberated" British South Africa, gaining several valuable port cities as well as at least one source of Coal.... Again, you can read up on that in the story thread. While some of the strategies I used are pretty much specific to AoI, most can be used in ANY game or scenario (I used the Artillery SoD strategy to pwn half of Europe in the Napoleonic Europe conquest as PRUSSIA!

By the time that game was over and done with, I'd taken over Denmark, most of France, Sweden, all but two cities in Russia, two cities in England, one of which was London, and half of Italy.)