The cheapest way to get bitcoins is from an exchange like MtGox. You can also buy them from someone that already has them using a site like LocalBitcoins.com. To buy bitcoins from an exchange you have to be able to get money to the exchange. This can take time, even days, depending on the exchange and method used. However, there is a site called BitInstant.com which provides a way to get money to an exchange in about 30 minutes, or send the coins to a wallet address or email directly.
More ways to get bitcoins are becoming available all the time, but for most people the following steps may be fastest:
1. Create a free wallet at Blockchain.info (this method advised for small amounts)
2. Go to BitInstant.com and in the 'Pay From' field select 'Cash deposit', and in the 'Pay to' field select 'Bitcoin Address'.
3. Fill in your bitcoin address you generate from Blockchain.info; enter the amount of USD you want to deposit and the other information.
4. Click 'Send Funds' and follow the instructions
5. You should have your bitcoins in about 30 minutes!
You can also sell goods or services for bitcoins, or mine them. Once you're ready to receive bitcoins learn how to store them safely.