When I go to search, it says, "Search iPhone" in the box.
How do I change that to search Google?
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If you type in the item you want to search for, an option will appear "Search Web". Press this and it will do a Google Search for the item you entered.
You cannot change the text itself (but the above should suffice in terms of searching google for a search term).
Matthew's answer above re the mail notifications is correct.
Also consider the excellent Google Search app.