Hmm, I'll give it a try. I have answered many questions on this forum, so I should have some tips.

Always nice to get a free game.
1) After settling your capital, develop the technologies necessary to improve the special resources around you capital and build the workers necessary to build the improvements.
2) Try to improve every tile that is being used by the citizens in your cities. An improved tile is at least twice as productive as un unimproved tile.
3) Manage your workers yourself. You won't learn the game by automating them and after a little while you're a lot better than the Artificial Intelligence (AI) at managing them.
4) Once you have settled, explore the region surrounding your capital with the early military units or scouts to find good spots for settling new cities and to find interesting resources for your civilization. The goody huts in the land can also help you in the beginning allthough they can also have negative results.
5) Most units need some special resources to be build. Copper (visible with bronze working), horses (visible with animal husbandry) and iron (visible with iron working) are the early resources that you will need to find and connect to your cities to build stronger units.
6) Bronze working is a very valuable technology after the very early game. It allows you to chop down forests for extra production in your cities and so that you can build other terrain improvements on that spot, it shows you where the copper deposits are located on the map so that you can build axemen and spearmen and it allows the valuable slavery civic.
7) The slavery civic allows a new form of production where you can sacrifice population for production. With granaries to regrow the population quickly, this form of production can be very powerful, especially in cities which don't have a high production but do have a high food output to regrow the lost population.
8) Barbarians and barbarian cities spawn randomly in the fog of war. Place a few units on defensive positions around your cities like forests and hills to push back the fog of war. This way, you'll get less barbarians and an early warning force for approarching barbarians.
9) As your civilization expands, your upkeep increases. Try to develop your economy while expanding by building the cottage terrain improvement and by building markets and courthouses in your cities. If your upkeep reaches 70-80% of your total commerce production, you should consider temporarily stopping the expansion as you don't want to go bankrupt.
10) Big cities are more productive and have better economies but they are also more unhealty and less content. Try to combat this unhealthiness and unhappiness by connecting food and luxury resources to your cities by connecting them with the required terrain improvement and roads or by trading for them with other civilizations.
11) If you attack cities, you'll need a significant attack force that is at least two to three times as big as the number of defenders. If the city has a (significant) defensive bonus, you'll also want to have catapults to bombard the defensive bonus away and use the collateral damage these units can inflict when they attack directly.
12) Don't let defending units heal inbetween your attacks; attack with a large enough number of units to completely defeat the defenders in one attack.
13) Get multiple defenders in cities that could be attacked by barbarians or other civilizations. Cities that aren't near the borders can make do with a single unit to reduce unit upkeep.
14) The computer opponents like to attack a civilization which looks weak. If you build very few military units and are near aggressive leaders (like Montezuma), then don't be surprised if they suddenly attack you without provocation.
15) If you don't understand a certain detail of this game, then become a member of Civilization Fanatics Center and ask your question there. There is even a special section for newbie questions and they will treat your questions nicely there even if they are very basic.