The easiest ways to learn to play Civ:
- Just jump in: Set the game on the easiest setting, set it for a small number of opponents, and play. Listen to the in game advice.
- Find a few "Let's Play" videos on Civilization 5
- Find a strategy guide or FAQ on the game and read it.
Civilization 5 is a 4-x wargame. This means that it is expected that you will need to: eXplore, eXpand, eXploit, and then eXterminate! So the game is built around the phases of exploring the map, expanding (building new cities) in the better available areas of the explored map, maximizing resources worked by the cities, and waging successful war on all that are smaller than you.
If you want to take a slightly more peaceful path to victory, you will still need to build up your military (to keep the AI from deciding you are small enough to knock off), and then go for the science victory (build a rocketship to start a colony on Alpha Centauri) or a diplomat victory (be so well liked by all the AI that they vote you president of the world for life).
This should hopefully be enough info to get you started. For more specific strategies, I'd suggest consulting strategy guides.