I think that you should firstly check out the map of the rail system:
http://www.nationalrail.co.uk/static/images/structure/css/nationalrailnetworkmap.pdf
Please note that the lines are not as straight as they appear on the map. Note that going into the south-west that you have one main line west of Exeter with various spurs coming off it. People with cars would find it easy to reach the places on the coast that the railway does not reach - such as Clovelly. You can find buses here:> www.traveline.info
The Britrail Pass may not be good value for money if just making short hops. Check out times & fares here:> www.nationalrail.co.uk
Note that longer journeys have cheap advance fares for specific trains. These ‘cheap’ fares are loaded about 11 weeks ahead and price gradually rises - though even pre-booking the day before can save money.
I would not bother with Weston. The Severn Tunnel (between Bristol & Newport for Cardiff) is currently closed until July 9th 2018 for electrification and passengers are being bussed over the Severn Bridge. (London to south Wales trains are being diverted via Gloucester). Cardiff does offer several nearby places of interest that can easily be reached by public transport including St.Fagans Castle & Museum, Castle Coch, Caerphilly Castle, Dyffryn Gardens (requires 1 mile walk from village of St.Nicholas). The city centre does of course have Cardiff Castle and 1 mile to the south is Cardiff Bay - not a beach but a vast harbour with waterside restaurants etc. To see all that lot, you would need to stay in Cardiff for maybe 3 or even 4 nights.
You may also like to consider going to north Wales by train. The line from Shrewsbury (worth a look) to the west coast is very scenic. Porthmadog* would make a good base for the Snowdonia National Park. For bus travel in Wales - try this site:> https://www.traveline.cymru
- 2 narrow gauge steam railways go from Porthmadog into spectacular scenery. www.festrail.co.uk