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"Change Passphrase..." obviously lets me encrypt the wallet with a new string of text. I think (correct me if I'm wrong) that this creates new private key(s) inside wallet.dat.
How are existing addresses affected when the passphrase is changed?
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The wallet actually uses a randomly generated master key to encrypt the keys, and this master key is encrypted using your passphrase.
When you change the passphrase, all that changes is the encrypted master key. The actually encrypted wallet keys aren't touched at all.
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If your wallet hasn't been compromised (how can you be sure?) then it should be enough to secure it with a strong passphrase and then be sure to back it up.
If your wallet has been or even just might have been compromised then securing it with a strong passphrase now might not be enough. Sending the Bitcoins at risk to a new address in the same wallet generated after the strong passphrase is put into place might not even be enough. It might take creating a whole new wallet on a known to be uncompromised system (how can you be sure?) and securing it with a strong passphrase before using an address generated in it to establish security.
Backing up your wallet to a cloud, e.g. emailing it to a Gmail account, etc., puts it at risk. If you ever did this before securing it with a strong passphrase then you should really move to a new wallet. Deleting the pre-strong passphrase copies is an illusion.
Randomly generated... So, is the strength of private keys dependent in any way on the strength of the master key? Will a master key passphrase = "Password1" result in strong or weak private keys? – Robert Claypool – 2013-05-19T20:11:31.387
2The private keys, and the master key, will be strong no matter what. However, an attacker with access to your wallet will attack the weakest link: the password you used to encrypt the master key. – Nick ODell – 2013-05-19T23:33:59.850