Play the demo when it comes out tomorrow. Until then, here's the manual:
http://www.civilization5.com/#/community/feature_manualCiv in a nutshell: build up your civilisation from a nomadic tribe to a globe-spanning modern superpower. 6000 (I think) years of gameplay controlling how you expand, what you focus on, what you research, what you build; all while interacting with the other civs around you. Win by various means: conquest, economic, cultural or launch a spaceship.
As for playing with friends: most of my Civ experience (II through to IV) has been singleplayer though I have played Civ IV multiplayer quite a bit but I've found SP to be more enjoyable as you can set the pace of the game yourself (you don't have to wait for your friend to finish his massively long and complex turn before you all move onto the next one).