Caveat: While you might use Google Apps to host your email [1], I would be leery of using Google to register the domain name that you use for this.
A major point of this exercise is being able to use control of the domain name registration to change its email support settings (change your email host), if/when needed.
You might find exercising control of a registration initiated through Google, and/or moving such a registration to a different registrar, difficult. And if so, you face Google's [cough] "legendary" customer support.
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[1] I believe that you can still host a single address for free; multiple addresses / users under an Apps account is now a paid-only feature.
A major point of this exercise is being able to use control of the domain name registration to change its email support settings (change your email host), if/when needed.
You might find exercising control of a registration initiated through Google, and/or moving such a registration to a different registrar, difficult. And if so, you face Google's [cough] "legendary" customer support.
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[1] I believe that you can still host a single address for free; multiple addresses / users under an Apps account is now a paid-only feature.