27 August is a Bank Holiday - so on this day and Sunday 26th - you can expect many shops in city centres to be closed. It is also likely that if the weather is good, roads to the coasts and inland attractions will be busy.
I was in Bath yesterday and parking can be very difficult. Most on street parking was limited to 2 hours - and you had to pay. (I used the bus Park & Ride from the edge of town).
Bath > Cardiff: the Severn Bridges still have tolls - I think it is £5.80 - driving into Wales only. Have your debit card handy and hold it over the scanner. The M4 around Newport can get jammed at busy times due to it being very busy and everyone having to squeeze through a 2 lane (each way) tunnel. You may like to consider crossing the Severn estuary on the M48 bridge and coming off at the first junction on the Welsh side and taking the A466 up the Wye Valley to see the ruins of Tintern Abbey (built in the 1100’s).
In Cardiff, I would not choose a hotel in the centre due to parking issues. Consider Holiday Inn Cardiff North or The Village Hotel -both are either side of the Coryton Interchange = M4 Junction 32. Just to the north of this interchange is Castell Coch. (It looks like Castell Coch is closed for long term repairs). If on leaving Castell Coch’s driveway you turn hard left, it is possible to follow backroads over the mountain to Caerphilly - which has a massive medieval castle - though the town is not up to much to put it mildly. From either of these hotels, it would be very easy to join the M4 going west and come off at the next junction (33) & take the A4232 for just over a mile - then turn into slip road for St.Fagans Castle & Museum - beware of sharp turn = 10mph. After St. Fagans, you may also like to consider visiting Dyffryn Gardens - which are about a 14 minute drive to the south west via the Culverhouse Cross interchange off the A4232.
Here is a map of the area mentioned above:>https://email@example.com,-3.2512436,15.7z/data=!5m1!1e1
You should be able to see both hotels. Now look for Longwood Drive. If going into the centre of Cardiff, I would suggest you drive down Longwood Drive towards the river. Here you can park the car on this quiet road for free - then take a short walk south along the path by the river and cross the footbridge to Radyr Station. From the ticket office, purchase a return ticket to Cardiff for £4 each. Trains are about every 10 minutes. If you see one parked at the station, this will follow a different route to Cardiff Central than the ones that pass through (but stop). If you want to go to Cardiff Bay (not beach), take the train to Cardiff Queen Street and change for the Bay shuttle. If doing this, I would suggest you pay 50p more and ask for a Cardiff Capital Day Ranger for £4.50. >https://www.arrivatrainswales.co.uk/rovers-and-rangers/
Although Swansea lies on a bay, it is not an attractive city due to being severely damaged in the war and then re-built in the 1950’s and 60’s. However, the coast of the Gower peninsula just to the west has spectacular beaches and cliff scenery. If you are going further west into Wales, you may like to consider skipping Swansea and Gower and head for Pembrokeshire. Tenby is a very attractive seaside town. Very attractive beaches and cliff scenery lie to the west of Tenby. See this map:> https://firstname.lastname@example.org,-4.4829901,50430m/data=!3m1!1e3!5m1!1e1
If not going west of Cardiff, consider taking the A470 north from Cardiff and up through the middle of Wales. Most of this route is scenic. 30 minutes north of Cardiff and you will be driving through the Brecon Beacons National Park.
Google the places mentioned.