Localbitcoins.com is a good starting point, but they're starting to verify the identity of users with large trading volumes or so I hear. You can try their competitors if that's a concern. You can't buy with a credit card on those platforms though. Still, peer-to-peer is your best bet.

Why would you want to use a credit card if you want to be completely anonymous anyway? You know your identity is right on the card right? Lol.