where are the bridle paths in swansea?

Buy an Ordnance Survey Map of the area - they will show you (shown as dashed green lines, footpaths are smaller dashed lines). If you need help deciphering a map (and they can be confusing) there are loads of places on hte internet that will tell you.

Buy a guide book from a book shop

Look on say, google maps, or multimap at their maps

Try asking in a mountain bike shop for them, or a horse equipment shop

Walk around the area looking for the signs.

Try going on some of the smaller local roads as well

Its going to be very difficult to explain otherwise where places are - far better to have someone show you or tell you the good ones face to face

look on O.S maps they should be marked enjoy your ride.