Welcome to kitchen ranges. There are a few of us on here who use them. Kitchen rangers can be a bit different to start and run then other types of stoves.
On a kitchen range with coal, close any damper that feeds air to above the firebox.
That's the secondary damper and it's not needed on a kitchen range when burning coal. If you open it, all your doing is cooling off the flue gasses with too much air and killing the draft. It does nothing to help get the coal burning - in fact it's just the opposite !!!!!!!!
You only need that upper damper for over-the-fire air if your burning large pieces of wood to help burn off smoke.
There should be a damper that feeds air to below the grates. That's your primary air. That should be open fully when starting the fire. And, do not ever close that primary damper fully, or you'll starve the fire.
Plus, you want the ash drawer door closed, or you won't be able to control the fire and it will burn up too quickly to get it started.
If you have a damper in the stove pipe (MPD = manual pipe damper), open it fully when starting and leave it open until the firebox is full of burning coal.
Once you light the fire, close the ash door and leave the primary damper open fully (with the secondary always closed), and the stove in direct mode - meaning open the damper that diverts heat to the oven when closed. It should be a door somewhere near where the base of the stove pipe attaches to the top of the stove. You want the flue gases to go directly from the firebox to the stove pipe while getting the fire started.
Once the firebox is full of burning coal, you can then close the primary damper down to about 10-15 % open and close that oven damper so that the heat will then have to travel around the oven and shed more heat into the room. And, you can now close the stove pipe damper (MPD) to about a 45 degree angle to help hold even more heat in the stove. If the fire dies down too much open the pipe damper and primary damper a bit more.
Over time, you'll see exactly how much the dampers need to be adjusted with your stove and chimney draft set up to get the most heat without over-heating the stove, or sending a lot of the heat up the chimney.
Fueling the firebox. Your waiting too long to add more coal. It's especially critical with the first few layers. When the coal stops snapping and popping, wait about 10 minutes then you can add another layer of coal. Don't bury the burning coals with too much. You should be able to see burning coal down between the pieces you just added. A good rule of thumb for the safe amount to add each time is, have each layer no deeper than the thickest piece of coal.
And, we love pictures. Please post more about your stove !!!!!