Is there a way to move an e-mail that has been labeled directly over to the left side as it comes in with the filter system? I guess I mean have it filtered directly over, not have it come into my Inbox first? I think Yahoo did this.

Yes, simply select "Skip Inbox" while creating a filter

Oh, I see it now. But is there any way on the left side to indicate that a new e-mail has arrived under that particular label?

Thank you ArpitNext.

Yes, skipping inbox doesn't label an e-mail as read, hence you will see number of new (unread) e-mails with that label, next to the label name, on the left.

Indeed, if you enable the "hide read labels" feature in labs you can easily see where you have pending messages.