I've written about what bitcoins are and how you use them is up to you, but if you are going to use them, you'll want to know how to get some.
You can start by either mining Bitcoins, or you can exchange something of value you have for Bitcoins. I can't recommend mining because of the hardware costs and the time you might have to wait to get any coins — but you could sell something in Bitcoins or trade some of your local currency.
How to Get Some
First, create a wallet. Either download a desktop client and let it download the blockchain to get up to date, or choose an in-browser client.
Generally, eWallet providers aren't recommended because they will also have access to all your funds. However, blockchain.info's wallet has an in-browser interface where the wallet is only decrypted in the browser, so you aren't trusting them with the contents of your wallet, just with the storage of the encrypted form of it.
_If you also use their phone app, the private keys can be secured with a secondary password that will be needed to send funds. _
Make sure you have backups of your accounts too: if any online provider goes away, you still want to be able to access your bitcoins! Blockchain.info will send copies of your encrypted wallet by email or to dropbox if you set it up.
Once you have an account, you have a wallet and therefore, have some receiving addresses, so you can set about getting some bitcoins. If you'd like a gentle introduction before putting down any of your own money, try out one of the services giving away a few (0.0001BTC) bitcoins for solving a captcha. The coins should appear in your wallet after a short delay (they may wait to group a number of payouts).
Happy that it works? Want enough bitcoins so that you can actually buy something? Then you'll need to find someone to trade with. Assuming that you don't know anyone that has bitcoins that you can swap for cash, then an exchange site is probably the next best bet.
Sending money to a trading site involves risk. Bitcoin trading is not a regulated industry. Any time that a trading site holds funds, they are not in your direct control. This means it is recommended to withdraw your funds back into your wallet/account once you have performed the trade.
Open an account with a bitcoin exchange. BitcoinCharts will give you an idea of the most active exchanges by currency, along with the current market value.
Send money to fund your account at the exchange of your choice. The speed of this transaction is determined by the funding method you choose.
Once your exchange account is funded, you can make an offer for Bitcoins. You can chose the exchange rate that you will accept, but if there is no one already offering to sell at the price you chose, then you may have to wait.
Once the currency exchange is made, send the bitcoins to one of the bitcoin addresses in your wallet so they are available for you to use.
And What Can I Do With Them?
While some people will buy bitcoins as a wildcard investment, it's purpose as a currency is for value transfer between two parties. In other words bitcoin is a currency, use it for buying and selling!
For instance, if you're worried about the UK's proposed "snoops charter", try buying some VPN access with some pseudonymous bitcoins and an anonymous email account. It's about 1.8BTC for a month or 6.8BTC for a year at the time of writing, and there's complete instructions for any computer, router or phone.