How to Move Folders from ‘On My Mac’ to Gmail IMAP – Mail.app Data Synchronization

One of the ways to do this is in the gmail settings. Now that Advanced IMAP controls graduated from labs, there's nothing extra to install. Go into the web interface for Gmail and go into settings. Go into the Labels tab and you should have a listing of all your labels with some options. On the far right side is a tick box that says "Show in IMAP" If this is off then Mail shouldn't be able to get at that folder.

Gmail has a guide on how actions on IMAP clients correspond to Gmail actions. In order to remove items from your inbox in Gmail, you will need to delete them from your inbox in Mail.app -- this will "archive" the message in Gmail rather than moving it to the trash. The trouble with this is that emails that you move to a folder in Mail.app probably won't show up in your Mail.app inbox until you refresh it, so it may take a few steps to move and archive a message. Unfortunately, Gmail behaves so differently from normal mail services that using it with a standard IMAP client tends to be a pain. However, the reference in the link above should help you figure out which actions in Mail.app correspond to various Gmail actions.