You can use an online exchange such as BitStamp for US dollars, as well as many others for dollars, and other currencies.
See the wiki for a full list of ways to buy bitcoins.
Brokerage services allow you to buy or sell directly from them instead of relying on a third party to offer the trade with the exchange just taking a cut. One such service is Coinbase.
Local / In Person
You can use tools such as localbitcoins.com to find people near you who will trade cash for bitcoins. (Note, however, that localbitcoins strongly encourages you to use their service despite the fact bitcoin does not require using a 3rd party to do in person transactions)
IRC and Remote Trades
Mining bitcoins is very technical, and not for many people. Unless you already have the hardware and technical skill to set it up I would not recommend mining as a way to get bitcoins.
I believe by far the best way to get bitcoins is to provide goods and services for bitcoin. This stimulates the bitcoin economy which in turn makes the bitcoins you receive more valuable.