One thing you should know about your cities, as a newbie to Civ: each population (citizen) eats 2 food. So you want to make sure you can feed all your citizens, because they will work the tiles around your city. The city tile gets worked for free, so every time you settle a city you will get your free city tile plus one more worked by your 1 citizen.
Extra food goes into a pool to generate more population, so you can work more tiles in your city radius. In Civ V, your citizens can work all tiles up to 3 hexes out from your city (maximum total of 36 tiles, plus the free city tile).
You need to build a worker for each city. This is different from a citizen; a worker is an actual unit you build and he moves around the map and can improve the tiles around your city by building farms, mines, lumbermills, and roads. Improving the tiles will get you more food, gold, or hammers depending on what you build in each tile.
There are special resource tiles on the map and you want to get as many of those in your city radius as you can. These are things like gems, gold, silver, deer, cattle, horses, iron, etc etc. Always use your worker to build improvements on these kind of tiles first, because they can give additional benefits.
There are also special resource tiles in the water/ocean (fish, whales, pearls, oil, etc). Your workers cannot improve these; you must build a work boat in one of your coastal cities and move it onto the special resource to improve these.
Also, you'll notice that only some hexes around your city are colored. This is your cultural border, and these are the only hexes you can improve with your workers. This border will grow over time as your city produces more culture, and you can also use your gold to purchase hexes one at a time. Buying land gets expensive, so mainly only buy hexes if there is a special resource that you want in that hex.
When you are ready for another city, you build a settler and send him out to a new location and settle the new city. You'll want to keep your settler safe from barbarians because he cannot fight, so you can clear the area with a scout or other military unit, or you can escort him with a fighting unit. Same with workers; if barbarians come near your workers, you'll want to move him inside a nearby city or run him out of range of the barbarian, or kill the barbarian if you have a fighting unit close enough.
These are some of the basic concepts you'll want to know since you've never played Civilization before.