Heathrow to Bath - a lot of people say go by National Express bus/coach which is cheaper than the trains BUT the service is only every other hour. By train, the services are normally every 30 minutes BUT you would first have to take a train (not Heathrow Express to London Paddington) to Hayes & Harlington (which is just a few minutes from Heathrow) and then a westbound stopping train to Reading for an express to Bath.
On day 3, you could go by train to Salisbury and then shuttle bus to Stonehenge before returning to Salisbury to see the medieval cathedral and then train back to Bath. Some people on here have recommended Mad Max tours from Bath to Cotswolds. Bath is a difficult to place to have a car so if you do hire a car here, make sure your hotel has guaranteed free parking. All roads in/out of Bath are busy and not super highways. The greatest concentration of attractive Cotswold villages lie well to the NE of Bath but are doable s a day trip form Bath. Castle Combe is the best of those nearby and can be reached via some narrow roads on the final bit.
The drive that you suggest from Bath via Chepstow (castle) and Tintern and then Raglan (castle) is very pleasant for the most part. The first village N of Chepstow on route A466 is St.Arvans. After driving through village - a narrow country lane gos off at first bend & signposted Wyndcliff. You soon come to a sharp bend with parking area on R. A 20 minute walk up a zig zag path through trees takes you to Eagle’s Nest viewpoint - well worth it if you are driving. (Yat Rock viewpoint is also worth visiting if you have time - see if you can find these places on Google Maps.
From Raglan (castle only reachable from eastbound lane of A40) - you may like consider looping back to go west bound A40 (look at road layout on Google maps as it is complicated unless you know) and then consider heading for Abergavenny & the Brecon area for an overnight - assuming you have a car. If doing that, you would use the A470 from Brecon to Cardiff & could make a detour to see Caerphilly Castle on the way. Of course, you could go straight to Cardiff from Raglan.
I would not have a hotel in Cardiff city centre if I have a car due to expensive parking and jams around the core area as the Council have made bus & cycle lanes as they have declared war on the car. All well and good to have a central hotel if you arrive by train from Bath. So, if arriving in Cardiff by car I suggest you choose a hotel on the edge of the city. The Village Hotel by M4 junction 32 would fit the bill as you could commute in to the centre by bus or train from Radyr station - which has trains about every 8 minutes taking only around 12 minutes to the centre of the city.
Look at Google maps & you should see Castell Coch just to the N of the Village Hotel - well worth seeing - as well as Radyr station which has a large free car park. You should certainly visit the Museum of Welsh History at St. Fagans and from J32 you just drive W to J33 & then A4232 southbound - look out for slip road into Museum about 2 miles S of M4 - SLOW DOWN in slip as turning off is SHARP. If not having a car and staying in city centre - use Cardiff Bus 32 for St.Fagans.
Day trip to Gower is perfectly viable and driving non stop, it takes just over an hour to reach the start of peninsula. Best advice is to head for SW tip at Rhosilli first & then work back along S coast. Roads can be narrow in places - check them out on Google streetview. Be sure to choose a good weather day for this trip as the area is exposed so keep your schedule of days flexible.
You could drive back to Windsor or use the trains with changes at Reading and Slough. For this longer one way train trip - it would be cheaper to pre-book a specific train Advance ticket.