I'm starting out on Illustrator for the first time and am trying to punch out this rectangle (in red) from a group of paths to clean up the edge of a design. Is this possible? And, if so, how?
While you can use pathfinder to cut your art. However, is often easier and better to use a clipping mask instead. A clipping mask does not cut your art it just hides a portion of it. Now, even pathfinder can do live cuts so that underlying art is intact.
The main benefit of a clipping mask is hat strokes need not be expanded, meaning that the art can be rescued even from final files. The downside is that you need 2 shapes because the clipping mask describes what to keep.
That is handled with the Pathfinder tools. With all elements ungrouped, expanded and selected, the rectangle can be used to clip off the overlapping pattern.
Try the different tools in the pallette to see what they do but Divide will separate everything at the join, you'll have to ungroup and delete the edge parts.