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If you perform a transaction using your own wallet, for which you control the private key, that transaction is now public knowledge on the blockchain.

If instead you perform the transaction using a cryptocurrency exchange it won't be public. However, you can't perform transactions if that exchange is down, making the argument that "bitcoin fixes this" a moot point.

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Bitcoins entire premise is around the movement of unspents. If you always use the same wallet for every transaction it is pretty easy to track who sent the money. If you instead are always sending your money to a newly derived wallet from your HD (Hierarchical Deterministic) root it does make it much more difficult to track what money was money being spent on something and what money is part of a persons total value

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