Bicycle Mechanics - How do you operate Shimano STI shifters?

This is my first mountain bike, a 1990's Trek 830 Antelope. Up to this point I've only worked on older road bikes with friction shifting. The rebuild is going well, but I am clueless how these Shimano STI shifters work with 4 levers. Would someone be so kind to fill me in why 4 levers and how do they work?
Many, many thanks.

One grey lever upshifts, the other grey lever downshifts. Push 'em.

The above advice is good. More specifically, the larger bottom lever increases cable tension (unless you have a bottom pull front derailleur, that would give higher gears in the front, lower gears in back), and the top leaver releases cable tension. Being indexed, cable adjustment is really twitchy, so you'll probably have to spend some time playing with the barrels to make them work in all gears. Make sure you're using shifter housing and don't have any kinks in the cable, or it'll never work properly.

the larger bottom lever increases cable tension (unless you have a bottom pull front derailleur, that would give higher gears in the front, lower gears in back), and the top leaver releases cable tension.

This is incorrect - all modern front derailleurs (since at least the ~early 80s, AFAIK) operate on a 'pull-cable-to-go-to-higher-gear/release-cable-to-go-to-lower-gear' system, regardless of top or bottom pull.

In general, rear derailleurs work in the opposite direction - pull the cable into an easier gear and release cable into harder gear.

Some rear derailleurs introduced in the mid 90s (RapidRise or "Low Normal") work opposite from that, but they are a little less common.

Thanks for the help! Rear derailleur and shifters work perfectly, but front is stuck because of a kink in the cable housing. Should be a straight-forward fix.

You'll figure it out in 2 minutes. I had the same concern with my new brifters on my road bike. I figured it out in 2 minutes. I love both types of shifters!

One grey lever upshifts, the other grey lever downshifts. Push 'em.

But don't push them unless you're pedaling.

But don't push them unless you're pedaling.

why not? it won't do anything bad to the drivetrain; the chain will just hop to wherever you shifted to the next time you start pedaling.

why not? it won't do anything bad to the drivetrain; the chain will just hop to wherever you shifted to the next time you start pedaling.

Safe to use the top button (release cable tension). Not so safe to pull the cables (bottom button.)

The Exage rear derailleur that might have come with these shifters had a spring loaded lever which allowed you to pull the cable even when the derailleur was jammed against the chain.

The front doesn't have such a lever, and in general most derailleurs don't have that device, and you could make the cable slip, damage the shifter internals, or bend the front derailleur cage, if you tried shifting more than one gear without the chain moving.

A technique to learn is how to change gears while you're stopped. You hold the front brake, push forward on the bars to lift the rear wheel, and spin the pedals around while you shift.