How can I get rid of "Outlook Today"?

If you go into Tools -> Options, click the Other tab, click Advanced Options, click Browse... where it says "Startup in this folder", you'll get a dialog where you can change the folder Outlook starts in. It's buried quite deep, so here's a picture:

There is also a KB article that gives steps for Outlook 2003 and Outlook 2007. (method 2). This is especially useful for when you place an item in the root of your mailbox by accident.