OS X Mail Mavericks 10.9 Gmail accounts NOT in sync

In Mail.app 9.3 (and possibly earler versions—I've never tried) this can now be done fairly easily—more or less, anyway:

1. In the mailbox list, beside "Smart Mailboxes", click "Show" then click "+" to create a new smart mailbox.

2. For "Important", select messages that match all of the criteria "Message is in mailbox Inbox" and "Message is in mailbox Important".

3. For "Starred", create a new smart mailbox that matches all of "Message is in mailbox Inbox", "Message is not in mailbox Important" and "Message is flagged":

4. …or, if you're like me and instead have Gmail show "All Starred", you can use the "Flagged" mailbox. This isn't quite the same as the (misnamed) "All Starred" section of Gmail's Priority Inbox view, which excludes starred items already shown in any of the sections further up the page (notably: threads which are both important and starred), so if you want to be really pedantic, create a new smart mailbox that matches "Message is Flagged" and "Message is not in mailbox Important" (and make sure that's the smart Important mailbox; select the regular Important mailbox and you'll unintentionally exclude important starred messages not in the inbox!)

5. For "Everything Else", create a new smart mailbox that matches all of "Message is in mailbox Inbox", "Message is not in mailbox Important" and "Message is not in mailbox Starred":

6. Finally, drag these new smart mailboxes to the favourites bar at the top of the window:

Caveats:

• Mail.app will only show one of these mailboxes at a time, whereas Gmail's smart inbox will show up to four different sections all at the same time.
• Because Gmail keeps track of labels and important status per-message (even though they are displayed as if they apply to the whole thread), you will occasionally see threads appearing in more than one of these smart mailboxes at the same time.
• No guarantees about what happens to important status when you start dragging messages to other mailboxes!

The preferred method for syncing your GMail account with an iOS device is by using Exchange with Google Sync. I could detail the directions here, but Google has a great writeup that is far more robust.

If you have multiple calendars you'll want to follow this guide to configure your device to access them as well.