Might be worth looking at the Wiltshire TRF public FB page and you can see what they get up to, you can also find advice on on bikes, kit etc. It's a bit dear to join up (£56.50 for first year) but you get lots of trail rides posted on the private Wiltshire TRF FB page suitable for all capabilities so you can learn your way around without getting into trouble. You can also ride along as a guest to see if it's what you want to do.
Regarding bikes you can ramble round the plains perimeter road on your normal street bike.
However I wouldn't recommend riding off road alone anywhere in case you have a problem.
