I am having issues trying to install a new, light switch/power outlet
that will control a new light in my attic.

Here is where I am at: I found a junction box in my attic that four
total wires in it: 1 black wire pigtail, 1 white wire pigtail, 1 red
wire pigtail, and 1 copper (bare) wire that is grounded to a grounding
screw. What I want to do is install a light that will run off a switch
that also has a power outlet that will not be controlled by the switch.
I bought electrical wire (I think it's called something like NW 14/3,
but I don't have it in front of me so that might be a little wrong)
which consists of 1 black, 1 white, 1 red, and 1 copper wire. The
light fixture has a brass and silver connection. The switch has 5
total points of connection. Looking at the switch/outlet from the
front, on the right side, there are three total connectin points. The
top one (marked common) and one right below it that is attached via a
copper plate and the third is completely at the bottom of the switch
(the ground). On the left, two connection points that are at the same
level as the two on the right.

Now, I thought I had this wired exactly as the diagram says, but
obviously, it's not right. The light doesn't work. When I test the
outlet, if I test the connection between one of the vertical plugs and
the ground, round, hole, there is current. If I test the other
vertical plug with the ground hole, there is current. If I test the
two vertical plugs together, nothing.

How do I need to wire this whole setup to get it to work properly (ie,
the light runs from the light switch portion, and the plug is always
on.)

Thank you for you help in advance.

You need to know what your existing wiring is.
It could be 2 circuits of 120V therefore 240V also. Or it could be part
of a 3 way switch setup, so power depends on some other switch. Or
someone who knows less than you did something weird!!

Normal setup is:
Bare or green is ground - safety!
White is neutral - tied to ground at the service panel but carries
current
Black and Red are power - many possiblilities...

If this confuses you I suggest buying a book or hiring a
handyman/electrician.

RickR