First you have to signup at Coinbase, verify the account and you are good to go.
Coinbase accepts credit card and bank transfer, while a company such as VirWox accepts Paypal, so the choice is yours. Also keep in mind that some of these transactions take a couple of hours the first time and others may take days; that is normal, do not panic.
Money transfer via the bank takes 4 to 5 business days. Keep this in mind if you opt for this method. Bank transfer is a very bad idea if you are buying Bitcoins when the value is high...the decrease in value can make you lose money. However, this doesn't matter depending on what you are planning to do with your Bitcoins once you have it.