The Trail Riders Fellowship promotes the non competitive riding of the byways and green lanes in the UK. You need to think of it as rambling or horse riding but using a motorcycle. Trail Riding is widely misunderstood by general members of the public but what you are actually doing is riding along a legal highway that has not been tarmacked or concreted, which is of course how all roads and highways used to be.

This does not mean that it can not be a challenge, locally ( meaning Wiltshire/Somerset ) most green lanes are OK, but travel over to the Brecon Beacons and you will be faced with rock strewn trails, steep climbs and drops and " interesting " river crossings. Over to the east in Essex you can come across thick sticky clay and sections where you come out the other end with pond weed draped over the bike. One time when I went ( foolishly ) out on my own I had to dig the bike out with the peak of my helmet, it was that or abandon the bike and walk home.

Your bike will need to be fully road legal, I would suggest a trail bike between 250 and 350cc as being ideal. You can ride trails on a 125, but it will struggle in some conditions, and I have ridden trails on a KTM600 but a lot of the time it was not  manoeuvrable enough ( and a bit of a pig to pick up! ).

Four strokes are preferable to two strokes, they generally have better traction on the slippery stuff, do not make such an intrusive noise, and do not leave a trail of blue smoke, all important considerations when you share the country side  with walkers etc.

You need to be happy to stop to show courtesy to other users of the byway, the PR is important, there is a huge amount of pressure to down grade byways to footpath status especially from the Ramblers organisation. Which I think is a great shame, byways only make up 10% or so of the  rights of way, leaving 90 % available to everybody else to use.