Day Trip from Cardiff by Public Transportation.

Not as easy as it used to be but Brecon fits the bill. The previous, and still shown on some websites, direct T4 bus no longer runs, you have to get a train to Merthyr Tydfil and catch it from there. And they seem to have failed to set up an integrated fare to account for this (unless Start knows better...). Train ride is 63 minutes from Cardiff Central, departures on the hour and also calls at Queen Street and Cathays stations a few minutes later depending onwhere you are staying. bus is another 44 minutes, walk out of station and a few minutes down the quiet narrow street to the bus station. Buses go at 18 past the hour so it's eactly 2 hours from Cardif Central to Brecon.

We were in Abergavenny last month and really enjoyed it- it's a 40 minute train ride from Cardiff and it's in the Brecon Beacons. It's a nice place on its own but you can also get the bus a very short ride away to Crickhowell which I might describe as an historic market town, and it's on a river. There is also some great walking nearby- we did a very nice walk up Sugarloaf in the Black Mountains just outside Abergavenny. I know this isn't really what you had in mind but we also loved nearby Blaenavon, which is an historic mining area, and you can not only see all the mining equipment and miners' cottages but also go down into a real former coal mine.

Long flight into Cardiff Airport:> Note that you have 2 ways of using public transport to get into central Cardiff and neither is good. Outside the terminal is a bus stop > if a First Bus 304 shows up (usually at x27 minutes hourly), it will take a whopping 1 hour 30 minutes to reach the city center via a circuitous route. The other bus 905 is run by Adventure (departs at x35 usually) as a 10 minute ride to Rhoose train station. The train arrives about 10 minutes later & takes 30 minute to Cardiff Central. The trains are hourly but on Sundays - every other hour. You might wish to consider taking a taxi.

Note that on Sundays, no buses run to St.Fagans. Cardiff Bus 32 runs the service right into the museum car park but note that the last bus back might be at 3pm - so best to go early as it is quite a big site. The castle is undergoing repairs but the gardens are well worth seeing in that part of the grounds.

You might wish to consider visiting Dyffryn Gardens (NT) - use First Bus X2 as far as St. Nicholas (tell the driver to let you know - it will be the first village west of the city). Walk 1 mile down a country lane to the gardens. On the way, you will see a sign for Tinkinswood Neolithic Burial Chamber & this is about a 8 minute walk across 2 fields should you wish to visit. St.Lythans Burial Chamber can be found by exiting Dyffryn Gardens - turn R and L at the next junction and it will be a field on R just around a corner. This would add about an extra 20 minute walk each way from the gardens entrance so might be overdoing it.
https://bustimes.org/services/x2-porthcawl-john-street-customhouse-street-jg

Note that you can also visit Bath quite easily be train and it takes just over 1 hour.

The waterfront at Cardiff - known as the Bay is also worth a look - Cardiff Bus 6 from the Bus station - which is right next to the Cardiff Central train station. Also consider a wander in Bute Park - which is to the NW side of Cardiff Castle. (A ferry runs down the river from here to the Bay).

Abergavenny has been mentioned (above) and the small town is a 15 minute walk from the station. If you decide to visit this town by train - use *tap in/out contactless on the scanners as that is much cheaper than buying tickets. I have climbed the Sugar Loaf Mountain and it is quite a hike! Note that if you are going by train from Cardiff to Snowdonia, that you will be on the line that goes through Abergavenny. * Don't forget to tap out at Abergavenny as there is no gate/barrier.

I did Brecon to South Wales (in that direction, as I had started from Abergavenny) two weeks ago on the hourly T6 bus to Neath then the train (in my case to Bridgend) but all trains continue to Cardiff.
It is 40 minutes by train Cardiff to Neath at 50 minutes past the hour- total journey time 2 hours 15 minutes.

There is no integrated ticket via Merthyr, but there is via Neath- £23 day return (includes the bus, and gets normal railcard discount).

It is actually cheaper to buy train tickets (£14.70 Off Peak Day Return) and a Traws Cymru day ticket separately for £7.

I had a Powys Rover Ticket for £9 on the bus as the X43/43 isn't Traws Cymru.

In spite of it's name the Powys Rover is a wonderful bus ticket- it gets you to Abergavenny, Hereford, Swansea, Llandovery, Newtown, Oswestry (a large part of the way to Chester) and Bangor in North Wales, so it can actually (eventually, slowly) get you to Snowdonia.

Via Abergavenny is about the same travel time- 45 minutes on the train, say 15 minutes to walk to the Bus Station (although every other bus stops at the Station Road End) then 45 or 65 minutes on the X43/43 bus.

You can do a good circular trip by going by train to Merthyr, a Powys Rover ticket for the T4 and T6 buses), then return via Neath.

Abergavenny St Mary's Church is also known as Abergavenny Priory- being a real ex former Benedictine Priory and is said to be the biggest Parish Church in Wales- quite an interesting Church and very friendly docents.

PS- Crickhowell has some very disruptive road works currently, supposed to be finished on 19 June- the town is not looking it's best with queues of traffic everywhere trying to get through pretty hopeless temporary traffic signals.