Hey all,
I've had a number of successful games on Regent, and am finishing off a victory on Monarch (admittedly, Monarch has also soundly trounced me several times, so I'm no master). Let me see if I can help out with a few things:
First off, expansion. Clearly, the AI is programmed to do a damn good impression of a rabbit on viagra. On Regent and below, you can keep pace, and even out-expand them. Above regent, it gets pretty tough to pull that off.
Choose a civ based upon your style of play. If you like to play a mostly peaceful, builder-style game, I strongly suggest the Babylonians. If you like to be aggressive, I suggest the Aztecs or Japanese... or even the Iroquois (NASTY special unit).
Trading with the AI is CRUCIAL. This means several things. Early on, your science rate is terrible because you have very little commerce. There is a cap, however, on how long tech research takes - no longer than 32 turns. Thus, early on I would suggest putting your science rate down low (like 20%) while you are just punching out settlers. This, coupled with money you will get from huts and dispersing barbarian settlements, will allow you to buy tech from AI civs. This will generally cost you 30-100 gold per tech. Obviously, you can trade your starting techs (usually the AI will want some gold, too, and even though it doesn't seem fair, just roll with it).
Trading with the AI also means trading resources. I am generally very careful about trading strategic resources, but luxury resouces are another matter. You can get all sorts of stuff out of the AI for luxuries. Tech? Sure. And the deal lasts 20 turns, after which you can get more (if you're still behind, for some reason). ALWAYS renegotiate after 20 turns! You will notice the AI does. They will also pay handsomely for luxuries. The larger the AI civ, the more valueable lux. resources are to them and the more they will be willing (and able) to pay. This should help any money woes you have, and hurt the AI's budget.
I usually switch to republic asap, and when I do, I'm often losing money. The best way to counteract this, from an internal standpoint, is to develop currency and build marketplaces. Banking is on the way to democracy, so go for that too and build some banks. Once you've build 5 or 6 banks, you can build Wall Street, which I HIGHLY recommend, because it will give you 5% interest on your treasury, up to a max of 50 gold per turn.
The forbidden palace is another essential thing. If you're lucky enough to have a leader, I would suggest using him to build this small wonder in a city far from your capitol that is centrally located with lots of other 99% corrupt cities around it. Those cities will suddenly be productive, and you will find tech research will speed up and your finances will improve. If you don't have a leader, build the forbidden in a city with enough production that the wonder doesn't take forever, but that is located in as good a spot as you can get for corruption reduction (this can be tough, I've spent time agonizing over where to put the FP).
Explore, explore, explore! If you're playing on continental or archipelago (sp?) maps, this means building a galley or two and trying to find the other continent(s). Once you do, and once people discover navigation and have harbors, you can trade with the civs "over there" as well, which should help you even more. I often will trade strategic resources to civs on other continents, as I'm less worried about them building troops and using them on me.
As for fighting, the AI is more challenging early on. I usually fight one ancient war and then hunker down, work on my empire, trade, etc. until the industrial (or even modern) age before fighting again - unless an AI civ picks a fight. The AI may well pick a fight if it thinks you're weak. This means it simply compares how many units it has vs. how many you have, and if it think it has enough, it may start demanding things or just flat out attack you. This is why it is a good idea to maintain a solid army. I know it costs money, but believe me, it can be done if you build up your infrastructure and use the money-friendly governments (republic, then democracy).
Anyway, I could probably go on for a while, but I'm starting to lose focus... so I'll stop here. I hope this helps ya out.
-Arrian