Directly from Hendre Rhys Gethin, you are spoilt for choice for mountain biking trails. The renowned Gwydir Mawr (formerly Marin trail) is a 25km loop blasting through Gwydir forest. The start and finish is a 20 minutes bike ride away, however, the loop is accessible at various points from Betws y Coed.
Penmachno is another fun and free mountain biking location, which can be ridden to from Hendre Rhys Gethin. It consists of two tracks Dolen Machno and Dolen Eryri, or for the adventurous, both can be combined for an epic 30km ride.
Both of the above sites are not for profit and are maintained by volunteers, therefore, if you can, please give generously at the donation box to ensure the upkeep of these trails.
Slightly further afield, Antur Stiniog offers some of the best mountain biking facilities in the UK with an uplift service to 14 downhill trails ranging from green to black.
If you manage to exhaust all the local trails, Coed y Brenin and Coed Llandegla are approximately 50 minutes drive away from Hendre Rhys Gethin. Both offer an excellent range of trails to suit all levels and have an excellent range of facilities including purpose-built centres with bike shop and café.