I'm looking to challenge myself more than just running and wanted to know if there were any specifics in how to begin trail / fell running (In this case running up hills / mountains).
Is there anything specific about becoming a fell runner which I should know about, or is it simply "Find an elevated trail and run it" ?
A few random tips from many years of mountain running:
In general there is nothing better than what you said: Go select a hill and start running. If you have (possibly experienced) company all the better. There is one point that you should keep in mind though: Running technique.
On flat ground technique is already important for efficiency and to go easy on the joints. This is even more important when running down. You need to take care to always run on your forefeet and make ground contact with legs slightly bent, otherwise knee and/or hip problems will ensue. This is especially difficult, as one usually starts a run at the lowest point, so in the end there is a descent. Therefore you must always reach the top with energy reserves for the final descent. For me this is very difficult, as a "summit" is always a great motivation, I need to actively break me down. Another option is to run from a high point or to a place where there is some sort of transportation down.
Adding on to the other excellent answers, here would be a few of my additional tips:
The basis is indeed "find a trail and run it". Or for fell running "found a summit and run up and down it". I like to stress this first of all, because all the well-meant (and very sound!) advice often makes it sound harder than it really is.
In addition to the other great answers, which brought you a lot of very useful information, I'd like to add some more specific English aspects, because that is where your profile puts you and because you also asked about fell running.
Tie your shoes. And tie them well. In the UK you have a lot of mud. Sometimes liquid mud, sometimes sticky mud. You'll learn to be an expert in the different kinds of mud, but often it is hard to tell the kind before you actually step in. At one point I sank waist deep while running in the Peak District. You don't want to dig out a shoe at that point.
Especially in the mountains, learn to navigate. You have lots of clouds and fog and grey weather. You can get really lost in thick fog. And descending in the wrong valley can easily add a few hours to your trip.
Look for the grassy lines when descending. Uphill the shortest way is usually the fastest. Downhill it is usually a grassy line, even if it is a lot longer and not following any path. Descending on grass you can really forget all breaks and feel the childlike feeling of yieeehaaaa. It is an amazing feeling. If you fall you were going too fast. If you don't fall you can still go faster.
At first, stick to paths or close to them, also on access land. When running off-trail it is easy to misjudge the ground underfoot and to loose your bearings. Once you start to know the trails a bit and have some experience there, head for the other parts. Running in that boggy, swampy, marshy ground that the UK is so fond of, can be painfully slow.
Besides the good answers above, think about your breathing technique and step length.
First, adjust your breathing, when running uphill short distances you can increase your breathing speed (e.g. once per 2 steps instead of 3 or 4). Don't do this for too much distance. If the distance going uphill is too far:
Second, adjust your step length... you still can use the same pace, but make shorter steps, so short, you run uphill with the same amount of energy as you would go on a normal slope. In this case, don't change your breathing speed. When you are adjusted to this, you can increase your step length slightly.
