I've seen plenty of people fail to light coal efficiently and the same applies to other fires. Since I'd got the materials, I thought I'd share some fire-lighting experience.
Lighting fires is a much less common task for the average person these days, and if you stuff it up you don't impress. Light it first time and you demonstrate that you have mastered fire.
Step 1: Let's clean it up.
Old ash and cinders will restrict air-flow, this makes for poor-burning. In addition, having ash up against the fire-bars can cause them to overheat due to lack of sufficient air-flow, they sag and "burn through".
Rake the remains of the last fire such that ash falls through the grate and pick-off the cinders for re-use. These are the lightweight dark lumps, not powdery un-burnable pieces of roasted shale. Clear the fire-bars of small cinders, clear all the ash.
You are off to a bad start if you don't do this
Step 2: Build your fire - paper
The paper should cover your grate, but with plenty of space to allow air-flow.
Don't go above one layer, as the paper burns down everything on top will drop, leave it at a couple of inches, no more.
The purpose of paper is to ignite the wood (next), you need enough, but too much will clog the fire-bars and cause stack-collapse problems.
If you find your paper doesn't burn well, stuff a loose sheet under the grate and light it. Keep stuffing sheets underneath and burning them, occasionally breaking the ash up with a poker.
Step 3: Build your fire - wood
Layer small pieces of wood (kindling) alternately such that you form a "raft". The construction should be a bit like a wooden-pallet, it is there to support the coal and ignite it as it burns. Criss-cross the wood so that it is in some way a structure and cohesive. When your paper is gone you want the wood to hold it's position rather than fall apart.
Choose a mixture of thick and thin. Thin will burn easily and produce heat, thick will sustain your fire and ignite the coal.
As a rough guide, aim for a cross-sectional area of about 1 sq inch maximum, but don't have the majority of your wood thinner than this.
What you see here is a bit more than "enough" - but it lit just fine. Not enough and you risk having to start again.
*maybe a massive amount of paper, stacked really high, someone show me in a video this if it can be done.
Step 4: Build your fire - coal
Choose pieces that are roughly the same volume as a golf-ball - too small and you have durchfall and poor air-flow. Choose pieces that are too big and they don't get enough heat from the wood to ignite properly.
Step 5: Light it!
If you've built this correctly all you have to do now is leave it for 30 min.
Coal needs time, the fire will blaze nicely while there's wood and paper left, but all that cellulose-fuel needs to heat the coals enough that when it's gone the coal-fire is self-sustaining. Coal produces gas and tar when heated only when it's "dried out" do you get the red-hot (or brighter) 99% carbon fire that makes coal so hot.
Step 6: Continuing care
Do not throw a bucket of coal on a fire, always put a bit at the edges, or in the middle. Picture 1 shows a strategic placement of coal at the rear of the fire, the others show the subsequent burn produced.
Step 7: Long-term maintenance
Ensure ash is removed from under the fire bars
When adding more coal do not tip a bucket-full on top. Add a little at the back, or the sides, or in the centre. Coal needs time to warm up, if you smother the fire with cold-coal you'll kill the lovely heat, and it will take longer to burn up. pile it up around the edges, when it starts burning: poke and rake it into the centre gradually.