Check out weusecoins.com. You will find on overview as well as show you how to get a wallet and some bitcoin to put in it. There are also some links to a couple of bitcoin markets.
Get a wallet
Smartphone:The android Bitcoin Wallet client by Andreas Schildbach seems very nice. It has a nice exchange rate feature. Allows you to back up your keys and seems to monitor the blockchain in a way that it isn’t downloaded to the device?
Computer: You can also download a client for your computer from weusecoins.com
Blockchain: You can get a wallet from blockchain.info/wallet. While you are there, take a look around to learn more about the system.
Get some bitcoin
Coinbase makes it easy to securely buy, use, and accept bitcoin currency.
I prefer to use expresscoin.com to exchange dollars to bitcoin.
Other bitcoin markets
gyft.com allows you to exchange bitcoin for giftcards of all sorts.
I plan to refine this to include a convenient current exchange calculator and also my plans to trade in precious metals such as silver in particular.