Remember, you do not have to purchase a full bitcoin. You can purchase a tenth of a bitcoin (0.1) or any other fraction. You can buy them online or in person. (We have more information about earning bitcoins or mining bitcoins).
To buy bitcoins:
CoinBase is an exchange based in the USA, but can be used from other places.
BitStamp is another exchange that is well regarded.
BTCChina is a an exchange based in China.
BTC-e is an exchange in Europe.
LocalBitcoins is a way to buy and sell bitcoins locally, in 192 countries around the world, including the United States.
A note regarding MtGox. MtGox is one of the oldest bitcoin exchanges, but has had issues in 2013 and 2014 which is the reason we do not recommend using them for purchasing bitcoins.