The first experience with a Civ game can be daunting. There are a lot of options, and not a lot of guidance on what might be the right thing to do. In the end, it's that freedom of choice that makes the game so interesting.
One thing that might help you is to have an idea of an overall strategy when you start a game. I usually base my broad strategy on the strengths of the particular leader. If you start out as Mongols, for example, your early strategy might be to expand quickly, converting as many barbarian villages into cities as you can. Then you'd probably want to research horseback riding as quickly as you can to take advantage of the Mongol mounted bonus. Of course as the game unfolds, your strategy will probably change based on what kind of cities you end up with and who your neighbors are.
The first order of business after founding your initial city is getting a unit or three out in the world to explore and fight barbarians. Exploring is important so you'll know where the good spots for cities are, and fighting barbarians is important for the early bonuses and gold you get.
For more details on building up each city, you have to look at the city's location and nearby resources to guide what you do. The first order of business for nearly every city is to get some kind of defense established. Otherwise the AI or a barbarian can just waltz in and take your city, and it can be hard to take back. After building at least a defensive unit (preferrably a defensive army), decide whether the city will focus on science, gold, production or culture.
A city with a lot of trade available (water and desert), or with a nearby trade resource like dye, can be used as a strong science or gold producing city. You typically only need one or two cities dedicated to gold, unless you're going for a gold victory or your strategy depends on rush buying a lot. Other trade-rich cities should be dedicated to science. In science cities you want to build libraries and eventually universities, while these buildings are wasted on gold cities, which use markets and banks to boost their output.
If your city has a lot of nearby forests, hills or even better (for later development) mountains, it's a good candidate for a production city that cranks out military units. In these cities you'll want to put up barracks, workshops, and eventually iron mines and factories so they can maximize their production. It's also handy if a production city is centrally located, or is only one or two road segments away from all your other cities. That helps you get units into the fight more easily.
Finally, cities that have an abundance of food resources, grassland, plains and perhaps fish and cows, are good candidates for cultural centers. Here is where you'll focus on temples and cathedrals, and city-growing buildings like granaries and aqueducts. If you can get a city like this on the border with another civilization, it's even better, because you can convert their cities with your culture.
City growth buildings, while especially important for culture cities, are really useful for all cities. The faster your cities grow, the more they'll be able to produce.
In terms of how to get the cities in the first place, there are several different approaches. Some people like to build a lot of settlers early. This is a strong approach for civs like China and Russia, since their cities tend to develop faster. Another approach is to wait until you get 100g from early barbarian village raids and then build another settler to go along with your free one. Any remaining cities you can get by conquest or conversion. Again, it depends on how you want to play the game. You may be in the mood for a peaceful expansion sort of game, or you may be feeling bloodthirsty.
There are no hard and fast rules for technology development either. Research the technologies that support your overall strategy. If you run into a hostile neighbor early on, you'll want to focus on technologies that help your military. If you have a little more breathing room, you want to try to get technologies that will grant you bonuses for all your cities. If you really have no idea, pick the cheapest one, since that tends to move you up the tech tree faster.
There are a ton of strategy articles here, and on the official 2K games site that can help you further. Also, there's an official strategy guide for the game which you may find helpful.
The most important thing is to have fun playing the game the way you want to play it. Everybody has a different style of play that suits them best. Trial and error is the best way to learn what works for you.
