Did you mean "bare" ground wire, not "black" ground wire?

If you have a white/black/red conductors then you seem to have the needed 3-conductor wire.

And the switch only has two connections on it? (not counting the ground wire).

I find it odd that somebody went to the trouble of putting in a 3-conductor wire but only using single-pole switches.

The ground wire is in fact surrounded by black plastic, just like a regular wire would be. However, it is obviously stripped at the tip and is going around the grounding screw.

Yes, the switch only has two connections. The black and red wire.

There is actually a bare grounding wire that is attached to another bare grounding wire. I attached an image showing the entire setup.

So what wire do I place where, if I get a 3 way switch?

I think some idiot replaced the three way switch with a two way (possibly the original 3-way wasn't grounded). The bare wires are the grounds. It is ILLEGAL to use black for the ground.

If there are indeed other grounds in the cable (bare), pig tail one of those out and use the black wire as the second traveller.

So, from the picture that I posted what wire am I suppose to use as the first traveler, second traveler, ground and common?

We are left to guess how this occurred. One is that "Mr Homeowner Previous" did this. At the least we must assume that he lived with this arrangement for some period of time-- perhaps as long as 19 years.
Installing the actual receptacles and switches on new installations is called "trim-out" and is often assigned to the newest and least experienced employees. And some companies pay for the trim-out by piece-work-- so much for each installed. The more installed per hour, the higher the effective pay rate. Has some advantages but is definitely not conducive to quality workmanship.
I suggest that you become very familiar with 3-way circuits and with the proper installation of equipment grounding conductors before personally doing any more to this situation.
Or hire a trusted electrician.

Here is a link that might be useful: Book

So then no feedback on which wire where then...

There is no way to tell which wire goes where until it is figured out where each wire comes from (power, the other switch, light) and which actually are the feed wires or travelers. Only an examination of the whole setup will determine what goes where.

If you can post a diagram of the wires coming into each junction box, both switches and the light fixture, we can probably tell you which is the most likely solution. There are several possible variations, otherwise.
Does the power supply cable come into the fixture first or to one of the switches? If to a switch, does the cable to the fixture come into that switch or the other?
With that diagram and a decent DIY book, you might also figure it out yourself.

What do things look like in the other switch box in the closet you are talking about?

Harder to see is what wires enter and exit the box at the light.

The simplest set-up for a home owner to deal with will have a single gang box with one non-dimming 3-way switch at each location, and just one (black/white/bare ground) cable and one (black/red/white/bare ground) cable in the box with the switch.

The cable that contains the red should go from box to box and makes up the travelers. The black and red ends of this cable should land on the screws on the switch that are the same color as each other (usually not black).

The other cable either comes from the panel (the feed in) or goes to the light (the feed out). Put the black from this cable under the unique colored screw (usually black and sometimes called the common or point). Splice together all of the white wires and use the bare ground to ground each other and the switch and the box if it's metal.

If you have a different wire arrangement than this, (and there are several that are both different and correct), and if you have only a small amount of experience then you would probably be well served by hiring a pro.

I personally can't recall ever seeing or installing a 3-way in a walk-in closet and the black conductor landed on the ground screw makes me think that you will end up hiring someone to sort this out. Good luck.

My guess is that the yoke of the new switch is energized depending on the position of the other switch in the loop. He installed single pole switches in place of old 3 ways and landed either the hot or traveler on the ground screw. Call an electrician. It won't take him long to fix this.

Is it possible that there is a back-stab at the left hand side of the switch picture? Is it also possible that the black wire attached to the ground screw is actually hot, or possibly just hot when the other switch is toggled? The other switch has the same black wire going to a ground lug?

There may be no way to fix this by simply re-wiring one switch. This switch is wired wrong and they may have made wiring changes in other areas to compensate. A decent electrician should have no problem figuring it out, but he/she will need to look at all the switches and lights before making any changes. The good news is that they apparently did use 3 wire cable, so the conversion to 3 way control can probably be done by swapping switches, without any other big modifications.

Bruce