Wild ponies do exist in Snowdonia but most of the Snowdonia Range are too steep & rocky to be a favoured habitat. The best place to see wild ponies in in the Brecon Beacons National Park (including the Black Mountains near the English border with south Wales).
Abergavenny would be your nearest railway station so, if coming from London (Paddington) or Reading via the Heathrow rail air bus link, you would change trains at Newport. www.nationalrail.co.uk
If coming from Shrewsbury, Hereford might be a more appropriate railhead for the northern Black Mountains area.
Getting from Abergavenny to some of the places listed below could be difficult without a car. I would only select for a pony trekking holiday those who could arrange transport from the nearest rail station. Also check out the local buses on http://www.traveline.cymru