Bath to Cardiff > well, you could go by train & it takes about an hour. If you go by car, you head N out of Bath on A46 to join M4 heading west. (M = Motorway = like an interstate). However, if going by car, you may like to cross the estuary on the M48 bridge (free) & come off at first junction on Welsh side to then drive up A466 to see the ruin of Tintern Abbey. You could also divert into Chepstow to see the Castle - but zoom down to check out the one way traffic system on this map >https://email@example.com,-2.8153295,11z/data=!5m1!1e1
Please note that the M4 frequently jams up at Newport due to a 2 lane each way tunnel at a very busy section of motorway.
I would suggest that you would need 3 nights in Cardiff in order to see everything in the area: > Cardiff Castle, Castell Coch, Caerphilly Castle & St.Fagans Castle Museum and the National Museum of Wales (good for art) in the city centre. All of these can be done by public transport from the centre. If you have a car, it might be best to have a hotel that is not in the centre due to parking issues. If using public transport, a hotel in the city centre is the best option.
It is possible to go by train all the way from Cardiff to Fishguard but the trains are infrequent. A train goes every other hour from Cardiff to Haverfordwest and I would consider this if you are not driving - but check for car rental as it might be more difficult in the far west of Wales. Fishguard is mainly a ferry port for Ireland and tourists might prefer to stay in St. David’s. Newport (Pembrokeshire) is a nice enough little place. The main ‘resort’ in Pembrokeshire is Tenby - on the south coast and is well worth a visit. Further up the coast from Newport, New Quay & Aberaeron are both attractive coastal towns.
You are going in November and weather can be rough - especially on the west coast if Atlantic storms come in. Gardens will usually be dormant at this time of year. However, you may wish to visit the National Botanic Garden of Wales which is just E of Carmarthen - should you be driving from Cardiff to west Wales.
Google the places mentioned.