The Borderlands Line between Bidston and Wrexham is well served by Arriva buses on which you can buy a £5 day ticket covering the whole of their Wales and North West areas. Buy it on the 407 bus which serves Bidston Station, the 418, 419, 437, 447 and 495 to Upton or the 486/7/8/9 group which links Liverpool, Birkenhead and Bebington with Neston.
For a great weekday out, use your Arriva bus ticket as above in conjunction with a day return rail ticket to Wrexham. This costs £4.20 from Bidston on all trains except the 0732 from Bidston on Mondays to Fridays, when it is £4.90.
From Wrexham you can use your Arriva bus ticket again on services as far as the Cambrian Coast. In particular, there is the Wrexham to Barmouth service, now upgraded as part of the TrawsCambria network with the number X94. This leaves from the bus station, and passes the Fire Station stop behind Wrexham Central. Unfortunately, with the exception (in theory) of the last two evening journeys in each direction, it runs straight past as a limited stop service.
Frustratingly, therefore, we have to advise you allow about ten minutes for the walk to the bus station from either Wrexham General or Central.
This JUST rules out the 0920 X94 from Wrexham as a connection for the 0817 from Bidston. So leaving Bidston instead at 0731 would leave you with plenty of time for coffee en route to the bus station as would the next buses for Barmouth at 1005 and 1120 with their connections leaving Bidston at 0817 and 0931 respectively. The journey to Barmouth takes about 2¼ hours and passes through Llangollen, Bala, Corwen and Dolgellau en route.
Return buses leave Barmouth hourly from 1550 to 1850, the last of which arrives in Wrexham just a few minutes before the final train of the day leaves for Bidston. This is ideal in theory, but risky in practice especially because some of the bus drivers don't realise that they are supposed to stop in Bradley Road behind Central Station.
It is always a good idea to tell the driver as you board the 1850 from Barmouth that you have a train to catch at 2127: even if running late, they will generally try to help you make your connection, even if it means stopping in Bradley Road!