Bitcoin is not completely anonymous as many early adopters believed. All bitcoin transactions are recorded forever in the bitcoin blockchain for anonymize bitcoin wallet to analyse. One problem that has been reported lately on reddit is that Coinbase is closing accounts when bitcoin bought from them or sent to them was used at a gambling website.
While most consider internet gambling a harmless pass time it is against the Coinbase terms of service. Not only gambling, but you cannot use bitcoin bought from Coinbase to buy cigarettes or use at an online pharmacy. In states where medical cannabis is legal and a dispensary accepts bitcoin a sure way to get your Coinbase account closed would be to buy legal cannabis with bitcoin you purchased at Coinbase. Coinbase is very good at what they do and while not much has been revealed about automatic screening systems they may have in place, it is clear that Coinbase is aware of the bitcoin wallet addresses of services on their prohibited list.
So How Do I Use Bitcoin Anonymously? I do not recommend you use the methods discussed here to evade Coinbase terms of service. I do however believe that people have a right to know how to use bitcoin anonymously. In order to do so you will have to learn how to break the transaction trail in the bitcoin blockchain so that any bitcoin you use can not be traced back to an address associated with your identity.
While I have not heard of bitmixer. Using exchanges to trade your bitcoin for an altcoin and back has also been recommended as a way to anonymize your bitcoin. I would like to outline another method that is simple and does not require you to open accounts at several bitcoin exchanges. Monero is a popular altcoin based on the cryptonote protocol that is recognized as the most anonymous cryptocoin available at present. The method I am outlining involves selling your bitcoin for Monero, then selling Monero to receive bitcoin back that has no connection at all to the original bitcoin. Create a new local bitcoin wallet on your laptop using a good wallet like Electrum or Multibit HD.