Seeing if the player is currently touching a specific part of the screen.

Right now if I want to make stuff happen when the fire button is pressed I use:

if (Input.GetButton("Fire1"))

What do I need do to make stuff happen when the player touches a certain part of the screen? I can store Rect information to define that area, but then what?..

Depending on your platform, you could use Input.mousePosition and/or the position property of each value in Input.touches. The script reference pages I’ve linked will provide you with more information about those values, but the short version is that they’ll give you a screen coordinate in pixels.

You could check that against one or more rectangles.

Many people find Camera.ScreenPointToRay() useful, for using screen positions with Physics.Raycast().