Depending on where you are in the world it can differ a little what is the easiest way to get started. I will point you in the direction of Poloniex, Bittrex, Kraken, or Litebit.eu. They will have instructions for you on how to get your first coins and of course you can buy them there and keep them on the exchange for further trading but you can also get yourself private blockchain wallets so that you do not have your money on an exchange but rather under your sole control.
Good luck mate!