There is no longer any rail connections in Newcastle Emlyn, the line disappeared under the infamous Beeching cull, but the legacy is a long walking area.
To get there you need to drive through the village of Henllan to the bottom, turn right and follow the road for a short while, there is a large lay-by where you can park just before the railway bridge. To get onto the track there is a small path to the right of the bridge or go under the bridge and there is an access a few metres along the road.
We drove to this today with the dogs and were pleased to find it quite empty of other walkers, although another couple and a dog appeared shortly after we got there. It was quite wet in patches but enjoyable enough and it was a lovely sunny, if crisp, morning.
We did this again on 19 November and mapped on Wikiloc. https://www.wikiloc.com/walking-trails/henllan-railway-track-43692969