When I push hard I get really painful leg burn almost straight away – so do I start with short, frequent intervals, on the road or offroad and do I start on gentle climbs or go for the big ones and ride it as fast and hard as I can?
I was in the same boat as you when i started riding, many moons ago! what worked for me was lactic threashold training. Once you have a good base, which it sounds like you have then you have to increase muscles tolerance to lactate, and train it to be used by your muscles. The way i did this was training at high intensity (just on the point of feeling your legs burn)but for times of around 1 hour to 2 1/2 hours. One of the least boring ways to do this is crit training or crit racing. Although there are no hills doing this, it is a sustained effort at near threashold, which it basically what you need to get up a hill quicker.
Eddy Merckx was from Belgium, pretty flat there, so hills are not compleatly necessary to improve climbing fitness.