Cry "Havoc!" and let slip the dogs of war. --WILLIAM SHAKESPEARE
What you'll first see depends on the interface you're using, but
typically there will be some sort of map or view of the world. Help is
available by typing ? or clicking on a button or menu item that
says `Help'. You can also just try to find your way around by
experimentation. (This is best done in a game by yourself, rather than
in one with a lot of other people.)
The game proceeds as a sequence of turns. During each turn, you and the other players get to move your units, which can be anything from cities to submarines to insects, depending on the game. In addition, there may be backdrop activities, such as changing seasons and weather, that go on all by themselves. These typically happen at the beginning or end of a turn.
Your exact goal depends on the game's scorekeepers. Most games have at least one, some have several, and some have none. There are many kinds of scorekeepers, so be sure you know and understand what they are before getting too far into a game! If there are no scorekeepers at all, the game is an "interactive experience" where you can do whatever you like. Any AIs playing in such a game will behave quite randomly.
A game may last anywhere from a few turns to many hundreds. Again, this may be limited by the game design, or perhaps by the nature of the game. For instance, a game of oil empires might be forced to end when the world's oil supplies are exhausted...