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.