You are lucky, Alwinton is a great place to set of from.
There are loads of good tracks and paths leading off from the village, and there are waymarked bike/horse and walking routes, e.g. the Border County ride is well signposted. Just set off with the map and you'll have a good run whichever way you go.

Usually http://walkingbritain.co.uk/index.php is a good place to find route ideas, though sod's law, nothing from Alwinton!