CEX.io is a heavyweight among Bitcoin companies. They not only serve consumers but also professionals in the Bitcoin industry and their deep involvement with the technological side of things speaks volumes about their operational security.
They do ask for some personal data in order to verify your good intentions but their security track record is solid and they sell bitcoins to international credit card holders, so they deserve to be our official recommendation if you want to buy bitcoins with your credit card from almost anywhere in the world.
Kraken.com is almost as old as the Bitcoin itself and they have captured the hearts of many Bitcoin enthusiasts over the years by providing an easy-to-use user interface and stellar customer support. You can send funds to a Kraken.com account from bank accounts all over the world, including the USA, making this platform a particularly flexible one.
It's really similar to setting up a Paypal account. Go to their website and sign up, link your bank account, make a deposit to your Kraken.com account (takes 3-5 business days), spend the bitcoins ;-)
Localbitcoins.com is an awesome peer-to-peer Bitcoin marketplace. It's the largest trading hub for everyone looking to either buy or sell bitcoins from or to other people, anywhere in the world. Like Ebay, just much less sucky.
The usability of their website is pragmatic and straight to the point – not ugly, not particularly beautiful either, but very practical and accessible. You can specify where in the world you want to conduct trade and all traders are being ranked and reviewed. A myriad of payment methods is available (cash, Paypal, international wire transfer, many more depending on your trade partners), and you can even use escrow services for added security.
All in all, Localbitcoins.com is an imensely powerful marketplace. As it is with many powerful tools, it's not the easiest thing to get into. We don't recommend it for beginners.