Right now the keyboard arrow keys are all mapped to "Horizontal" axis motion, causing my character to move forward/backward/right/left. I want to remap my arrow keys to be: (a)Right/Left == Turn right/left, and (b) Up/Down == Look up/down (or pitch the camera in 3rd person view).
How do you redefine which keys are associated with the "Horizontal" or "Vertical" axes?
For remapping the keys associated with the "Horizontal" or "Vertical" axes, you'd simply go into Edit / Project Settings / Input. So that's very easy.
However, what it seems you're trying to do is creating another behavior for those axes. In that case, you might have to change the controller you are using (e.g. the CharacterController). In that later case, you'd have to be more specific as to how exactly you have your scene set up. Changing the behavior of the axes is obviously much more involved as you may have to code the specific behavior you want (but that would also be a different question, I guess).
You can change all of this in the Input Manager.
http://unity3d.com/support/documentation/Components/class-InputManager.html