I am trying to figure out how to wire these with a new style light switches (the basic 1 and 2-way switches from Home Depot).
I've installed light switches before but this is confusing to me because these old switches really don't show what is what.
These are all independent of each other so they don't work off the same light.
The far left one is by itself which I can install myself with no issues.
Not sure what other information I can provide to help solve my issue.
Best Answer
The four switches have a set of "jumpers" that go from one to the next to carry the hot feed to the next switch. A simple switch has the always hot feed on one connection and the switched hot on the other connection. The switched hot goes to the hot side of the light and is hot only when the switch is ON (closed).
It looks like you could do the same thing with the new switch (the left one in the picture) by slipping the jumpers under the tab secured by the screw and tightening. So each hot connection would have two wires under the screw compression tab.
Another approach would be to use a set of four short pieces (black or red wires called "pigtails") and connect them in a wire nut to the hot feed (red wire). Then each pigtail would go to the hot side of one switch. In your case there would be five wires in one wire nut. That would take a fairly large wires nut. The sizes are color coded and a grey colored nut might be required to connect them. A red nut might work if the wires are #14.
EDIT
In the past the hot feed was carried to multiple switches by a single wire which went from one switch to another and looped around the screws. The insulation would be stripped at both ends and at each intermediate connection. The wire would normally be connected at one end to the hot feed with a wire nut.