I bought the game off steam about six months ago, and I've become addicted to the point where I've already played 200-something hours on it. I'm far from a top player of course, but it took me a while to understand certain concepts.
So, some things to consider: Make sure you understand the different things your cities produce. Commerce in particular took a bit of time to wrap my head around, because the tutorial and the encyclopedia aren't so clear about it: The coins you see on the tiles being worked by your cities aren't "gold" per se, but commerce (Commerce is also earned from trade routes for instance). The science slider at the top left of the screen determines how much of that commerce is dedicated to research. If you put it at 80%, that means that the remaining 20% of your commerce becomes gold that you accumulate in your treasury (unless you have the culture/espionage slider up, but you don't have to worry about those in the lower difficulties). However, some things in the game (such as the religious shrines you build with great prophets, or the priest/merchant specialists) give you gold directly instead of commerce. They have a different icon representing the two concepts: Commerce is a coin, gold is a mound of gold.
Basically, once you start having more cities, administrative costs start eating at your treasury. At this point you'll have to lower the science slider so that some of your commerce becomes gold that pays for the costs (see your financial advisor to know what those costs are). Don't worry about the slider too much (the game lowers it for you if you don't have any more money); what's important is the amount of beakers produced per turn (which you can see on the right of the slider). In other words, a civilization with 10 cities will have much more administrative costs than a civilisation with three cities, so it might have to lower the research slider to, say, 50%; however, with all the commerce that those cities might be producing (assuming they're well placed and everything), the 10-city civ might be producing 50% of (potential) 1000 beakers while the 3-city civ could be producing 90% of 400, so that the bigger civilization would still be gaining technologies faster. (another way of generating beakers, without using commerce at all in fact, is to run scientist specialists in your cities).
Speaking of which, research is important. Being in the lead technologically is always a good thing.
Food and Hammers are more straightforward, but just remember: you need food for your cities to grow. Every population in your city requires 2 food to survive, so considering that many tiles give less than that, your cities will typically not grow big enough to work all their tiles. And it'll be difficult to build many buildings or units if your cities aren't located near hills. Always consider which tiles can be worked by your city (I believe it shows you this when you select a settler; it's a kind of fat cross). After your city builds enough culture for its first "border pop", it'll get to work all of its available tiles, so if you want your city to be useful from the start, settle it next to the ressources you'll want it to work first.
I mentionned food earlier, and this leads also to an important distinction which I don't believe the tutorial goes into much: The ressource tiles give your city an added bonus (for instance, stone can help you build wonders faster, bronze lets you make axemen and spearmen, food ressources like corn give your city health) when there's a road connecting them to the city. However, it's important to remember that these ressource also increase the yield of a tile by a whole lot, regardless of roads. For instance, an irrigated (ie next to a river or a lake) rice tile, for instance, gives you a whole 5 food, which is enough to have two other citizens working tiles that yield no food, such as plains hills, which are excellent for hammers when they have a mine on them. Basically, you'll want to have all your cities (especially your early ones) next to a food ressource so they can grow and make themselves useful. Of course, for ressources like Copper, if you want to make axemen, you're going to want to link them to your city as soon as possible, but you can wait with ressources like corn and pigs, because your cities don't need the health bonuses until they're bigger and it's later in the game. Also note that as soon as you have the sailing technology, rivers and coasts essentially accomplish the same function as roads in that they can link ressources to cities and cities to each other to produce trade routes (don't worry about the trade routes: The game automatically chooses which cities to trade with based on which is most profitable, you can't even change that).
These are all things that I would have really liked the tutorial to explain properly, because my first games would have made a lot more sense to me. I'll suggest the same thing as above posters and start on the lower difficulty levels, but just for a few games, because they give you a lot of bonuses, and when you start playing at noble (the difficulty level which puts you on equal standing with the AI) or higher, you might have bad habits that simply won't work anymore.
Finally, a few tips:
-Always have an army, even if you're not planning on attacking anyone; on noble or higher, AI *will* declare on you if it feels like it, and all the wonders in the world won't prevent Montezuma or Gengis Khan from destroying your beautiful cities (this is something most people have to learn first hand, myself included).
-If you're trying to attack a city any later than 500BC, bring siege (catapults are available with the construction technology). Those can both reduce city defence and cause colateral damage (meaning that when you attack a unit, other units also take damage). They can't outright kill enemies if you're playing the Beyond the Sword expansion, though, so always have them accompanied. Also remember that attacking a city always puts you at a disadvantage, so you always want at least twice as many attackers as defenders (except when you're attacking spearmen with tanks, but even then...
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-Remember that with the Alphabet technology, you can trade technologies with the AI. That way, you can acquire Technologies that you didn't bother researching yourself.
-Religion affects diplomacy a lot. Having the same religion as an AI is one of the best ways to make friends, just as having a different religion can make you enemies (especially with leaders like Isabella, who is a complete religious nut).
-A lot of the leaders which have the "aggressive" trait are considered "warmongers" and can be dangerous to have as neighbours (Montezuma is particularly crazy). They can attack without provocation. In fact, a lot of AI will declare war on you even if they are "pleased" with you (AI attitudes toward you go thus: Furious, Annoyed, Cautious, Pleased, Friendly). It can be a good idea to make good friends with them (Friendly; pleased usually isn't enough) or to take them out before they become a threat.
Finally, I hope this isn't considered advertising or anything because that's certainly not what I'm trying to do, but the Civfanatics website (http://www.civfanatics.com/) has a ton of strategy articles, from beginner guides to extremely in-depth analysises of certain mechanics (check out the war academy section). The forums there are also great if you need help.
Hope you enjoy yourself as much as I have!
