Heathrow to Conwy has been explained in the England forum.
Please take a look at this map of long distance bus routes in Wales operated as Traws Cymru:>http://www.trawscymru.info/routes/
(The railways have been shown on blue). The Traws Cymru buses where the routes begin with a T are free to use on weekends.
To find any journey in Wales, use:>https://www.traveline.cymru
To get from Conwy to Aberystwyth by train, you would need to travel east back into England & get to Shrewsbury - where you change trains for one that will take you west to Aberystwyth. (Shrewsbury is worth a look / stop for lunch and take a later train if you don’t want to do it all in the 4 hours or so.https://www.visitshrewsbury.co.uk
Another way of doing it by train is to go from Conwy to Llandudno Junction (1 mile east) and take a train to Blaenau Ffestiniog where you change for the narrow gauge steam train of the Ffestiniog Railway - which will take you to Porthmadog. You would then need to walk up the main street to the ‘normal’ train station and take a train to Machynlleth and then change for Aberystwyth. Going this way is very scenic but you may have difficulty in getting the train connections to work reasonably.
Another way to do it is to go from Conwy to Bangor by train (or bus) and then take the T2 Traws Cymru bus to Aberystwyth - again, most of this route is scenic.
When in Aberystwyth, take a ride on this railway:> http://www.rheidolrailway.co.uk
From Aberystwyth to Lydstep, you will find some local operators link up to form a route with connections. Alternatively, T1 bus to Carmarthen and then train to Penally - ask accommodation provider to pick you up as Lydstep is very near. OR - T5 bus from Aberystwyth to Haverfordwest and then check out for the local connection on traveline-cymru.
Here is a satellite view of the Lydstep area:>https://firstname.lastname@example.org,-4.7635258,5905m/data=!3m1!1e3!5m1!1e1
You can see several sandy beaches and they are linked by a coastal path where you will see wonderful cliff formations. Be sure to go into Tenby and try and take a ferry over to Caldy Island (which I think is closed on weekends). Pembroke has a castle.
Bosherston - further west, has a lovely walk by lily pools down to the coast. If you do this & then turn E, you reach beautiful Barafundle Bay.http://www.visitpembrokeshire.com