In fact once someone has one of your Bitcoin addresses, they'll be able to see and track all transactions involving that address, including "shuffling" into new addresses. One can even construct a whole graph of transactions starting from some known bitcoin address since the chains of transactions are absolutely transparent for everyone. I guess it's a dream come true for the intelligence and data mining agencies.
And yes, the only way to opt out of that transparency is to use the laundry and mixing services - the chain of transactions ends once you put your sum into the service and retrieve them back - since you (most probably) will get other people's coins. Just be careful not to put too much since you may get back your own coins. Also, such mixing service may not be free.
This article discusses transparency and anonymity issues well:
https://en.bitcoin.it/wiki/Anonymity