If somebody does a post into a Forum that I am the moderator for but I see from reading the post content that it really should be in a different Forum, how do I delete it from my Forum and move it to the correct Forum? I also want to avoid the "This Post has been removed by the Moderator of this Forum" big red banner message.

<FORUM MODERATORS ONLY>

Open the post you want to move.  As a moderator, there is an "edit" option.  Clicking on that reveals a "move to" field which allows you to move the post to another Forum.

Ian