* To view only unread mail, use a search filter of is:unread (there are also a bunch of other tags you can use as well like is:replied or is:forwarded)

* To search for a substring (like Johns instead of Johnson), use asterisks as a wildcard: johns*

* To search only against a specific header from the basic search box, prepend the header name with a colon: from:william or from:will*

You can see the full list of search tips from the Zimbra Wiki.