you should make your bitcoin anonymous from the start. which means you should buy it with cash from someone and place it in a new address.
this way it will have no links to you, because buying from any sort of exchange or using credit card would link it to you.
I think this would be a good idea, if a large majority of people had bitcoin and you could find people to sell/buy anywhere in the world... but where I live, there is hardly no one that carries bitcoin or would want to sell them. I've check on localbitcoins website a good amount of times, and still have trouble finding someone in my area that's worth the drive.
Also, it would be nice to have bitcoinATM's to be able to buy with cash... but again, there are none where I live...
Depending on your location's AML/KYC policy, your ATM may need your driver's license in order for you to buy BTC. Some manufacturers, like Robocoin requires the operator to turn it on regardless of whether it is required or not. The best way is to buy a gift card and trade it on localbitcoins or the currency exchange section, just keep the receipt as some requires it to verify that you didn't buy it from stolen credit cards.