Post Cotswold suggestions without a car

Are you staying in the Cotswolds and, if so, where? There is a train from Oxford through to Moreton-on-Marsh; another option would be stay in or near Bath, which is a must if you haven't been there before. Direct links from MIM and Bath to London. There may be other train lines through the Cotswolds which I am unaware of, I will let others input on this. But generally besides those option you'd have to negotiate with the fickle and bizarrely timed bus timetables in more remote parts of the Cotswolds which is possible but requires further research depending on what exactly you'd want to see.
One place I'd recommend is Gloucester for its cathedral. Part of Harry Potter was filmed there and the building is free admission. Very interesting "coat hanger" arches and the buttresses are inside the building rather than outside.
There is also a lovely new shopping area by the canals
You can get to Gloucester from Oxford, but I'm not sure about from Moreton in Marsh.

The Cotswolds is a very large area, so it depends where you are staying. Clarity on that would be welcome. But Cardiff would be easy. Hourly train from Moreton in Marsh (if that is where you are staying) to Worcestershire Parkway, then hourly train to Cardiff.

But generally besides those option you'd have to negotiate with the fickle and bizarrely timed bus timetables in more remote parts of the Cotswolds which is possible but requires further research depending on what exactly you'd want to see.

The buses are not fickle and bizarrely timed. They run to specific timetables which have been enhanced several times this year, and on the trunk routes are now very good.
It is worth restating that the operation of all UK bus routes is closely monitored by the Traffic Commissioners who will and do take enforcement action if routes are not being run properly. That data is open access.
I or anyone else can access the data- although it's sheer volume can make interpretation hard.
Quite frankly the buses to the smaller villages are there to primarily suit the needs of the locals not the tourists. If all tourists were like most on this forum (and they aren't) then you couldn't run buses round their needs given the general mindset on this forum about public transport in the area.
Apart from fixed route services in the area there is also the wonderful Robin Demand Responsive service for local journeys to anywhere in the area.

You can get to Gloucester from Oxford, but I'm not sure about from Moreton in Marsh.

From Moreton in Marsh by train- as for Cardiff above, except Gloucester comes sooner.
By bus the hourly bus from Moreton to Cheltenham then the frequent (at least ten an hour) bus from Cheltenham to Gloucester,

If going from Moreton-in-Marsh to Bath, you need to change trains at Didcot Parkway. Much depends on where you are staying in The Cotswolds.

Gloucester has been mentioned and although it has a cathedral, I don’t think much of the city centre. A so called ‘outlet’ shopping area has been developed by the old docks but I would not go out of my way to visit.

As has been mentioned, you could go by train from MIM to Worcestershire Parkway from where direct trains go hourly to Cardiff. Alternatively, you could go by bus to Cheltenham from where direct trains go to Cardiff every half hour with the journey taking about 1 hour 10 minutes.

Did you know that 4 castles can be found in & around Cardiff?
https://www.cardiffcastle.com/the-castle/ is found in the centre and dates from Roman times. In the middle is a Norman keep (1087) but much of the castle dates from the 1800’s when it was owned by the Marquess of Bute.
CAERPHILLY CASTLE is a massive medieval fortress just a 20 minute local train ride north from Cardiff. The town of Caerphilly is a dump in an ex mining area but the castle is everything that you would expect of an 1100’s fortress complete with moat. If you are visiting Caerphilly, you might like to break the journey at Lisvane station and walk for 5 minutes across the car park and into CEFN-ONN-PARK (free) & nice place for linch left of entrance. In spring it is very colourful with azaleas, camellias, rhododendrons and bluebells all under the canopy of a temperate rain forest. https://www.outdoorcardiff.com/parks/parc-cefn-onn/
https://cadw.gov.wales/visit/places-to-visit/caerphilly-castle

CASTELL COCH stands on a hillside in north Cardiff. If visiting by bus from Cardiff - you need Stagecoach 132 from Greyfriars Road and alight at Tongwynlais (about 25 minutes out). The walk from the bus stop to castle is about 17 minutes uphill.
https://cadw.gov.wales/visit/places-to-visit/castell-coch

ST.FAGANS CASTLE & MUSEUM OF WELSH LIFE is just to the west of the city. The site is vast and needs at least half a day. You have grounds with buildings of significance brought from all over Wales and re-erected. The Castle is really a 1500’s Manor House. You will also find gardens near the castle. You will also find indoor galleries explaining about life in Wales through the ages - handy if it starts to rain. https://museum.wales/stfagans/about/ Cardiff Bus 32 but note that the last 32 bus back is about 3pm. Another company called First operates a number 320 bus service from Cardiff (Westgate Street) to St.Fagans (actually going beyond St.Fagans to Talbot Green). These buses stop outside the village gate to the St.Fagans Castle/Museum - so just press a button to open the gate & walk in. The 320 bus stop back to Cardiff is outside the village church. No 32/320 buses on Sundays.

LLANDAFF CATHEDRAL is about 2 miles NW of Cardiff Castle.https://www.llandaffcathedral.org.uk
If visiting Cardiff, I suggest you visit CARDIFF BAY - which is just over 1 mile south of the city centre.
https://www.visitwales.com/destinations/south-wales/cardiff/top-things-see-and-do-cardiff-bay
https/

THE NATIONAL MUSEUM OF WALES is found just N of the shopping area next to the City Hall. It has extensive art galleries.
If visiting Cardiff, note that the city holds major events - especially in the Principality Stadium. This can result in hotels charging massive rates to cash in on all the fans and transport systems are stretched.
Best to avoid such events if of no interest.

Is31nc I was only being hyperbolic about the Cotswold buses. I'm sorry if it was misleading!
My intended meaning was: check the timetables carefully. I have been stuck in remote areas of the Cotswolds before because I didn't look closely enough at the timetables. And in general the buses run less frequently than they do in other areas.

Thank you so much for the suggestions! We've been to Bath before and will be leaving from Moreton-in-Marsh. I'm guessing Cardiff would be a better place to visit as opposed to Oxford and Woodstock (Blenheim Castle) ? Is it best to purchase train/transportation tickets prior to arrival or wait until we get there?
Again, thanks so much!

Buy before hand.

If you buy on the day the fare is £77.

I'm looking at 6 January. Buy as separate tickets-
Moreton in Marsh to Worcestershire Parkway- Off Peak Single, and Worcestershire Parkway to Cardiff Central Advance Single (Train Specific).
You have 7 minutes to change from the High Level Platform to the Low Level Platform at Worcestershire Parkway, which is entirely adequate. If the train from Moreton arrives late at Worcestershire Parkway your ticket to Cardiff is automatically valid on the next train, 1 hour later.
On the 0918 from Moreton (arrives Cardiff 1125) you pay a total fare of £26.10. On the 1020 and 1120 trains a total of £31.20- so a huge £50 or so saving.

Buy a bit further in advance (up to 12 weeks) and the fare may be lower still.

We've been to Bath before and will be leaving from Moreton-in-Marsh. I'm guessing Cardiff would be a better place to visit as opposed to Oxford and Woodstock (Blenheim Castle) ?

Not necessarily so. The choice is yours.
There's plenty to see and do in Oxford. It's an easy and short train ride from Moreton-In-Marsh.
Plus, there's a coach (bus) directly from Oxford to Heathrow airport. Go early on your day of departure if you are departing late in the day. Go the night before and stay near the airport if you have an early flight home.

Blenheim Palace (there's no castle there) is easily accessed by taking a short bus ride (day trip) from Oxford.

If what you want are castles, go to Cardiff.

Cardiff v Oxford & Blenheim Palace is a difficult one. Oxford will be almost entirely old buildings - many belonging to the various colleges of the university. Blenheim Palace is well worth seeing. So, you have to weigh that against the places mentioned in and around Cardiff.

For convenience, Oxford & Blenheim Palace would win as they are very near to your base in MIM. You also have the advantage of the direct buses from Oxford back to Heathrow v the longer journey from Cardiff to Heathrow.

MIM to Oxford is £13.30 off peak return (pay on the day) and takes about 33 minutes by train.

What is little known on this forum is that you can very easily do Blenheim Palace as a day trip from Moreton in Marsh by bus (as from 31 March 2024).

Take the Pulhams #801 hourly to Chipping Norton (that's the bus from Cheltenham) then the #S3 hourly from Chipping Norton to Blenheim. Each bus ride will be £3 next year.
The S3 actually goes into central Oxford (George Street) after Blenheim so if going to Oxford either to stay or for the airports it may be more convenient than the train- as Oxford Station is on the edge of town.
George Street is just round the corner from Gloucester Green Bus Station for the buses to Heathrow and Gatwick, and the Oxford Tube bus to London.

From Cardiff to Heathrow there is also an almost hourly direct National Express bus 24/7 - it takes 3 hours- not much longer than the train (and a lot cheaper). That avoids at least 1, (or 2 or 3 depending on route) transfers on the train. And your luggage goes in the hold of the bus, so that bit is also easier than taking the train.

Don't do bus to Cheltenham then train to Cardiff, as that would involve the transfer from Cheltenham Bus to Rail Station- train all the way is far easier.

There is also a castle in Oxford- https://www.oxfordcastleandprison.co.uk/

Oxford is close. Also, consider Bath.
Cardiff is good, but you would miss some great castles not having a car.

All 4 castles in the Cardiff area can be reached by public transport. Frequent (like every 15 minutes) trains to Caerphilly; Buses > Cardiff Bus 32 or First Bus 320 to *St.Fagans and Stagecoach Bus 132 to Tongwynlais for Castell Coch - but that does involve a hike uphill for about 18 minutes to the castle. Cardiff Castle is of course in the city centre. A taxi from Caerphilly Castle to Castell Coch would help reduce time to travel between these two castles.

*Note that the 32 goes into St.Fagans Castle Museum car park but the 320 stops in the village by the church. You would then press a button on an electronic gate to enter (free). Best to tell bus drivers where you are going so they can tell you where to get off.

Thank you so much for all the great information! I really appreciate the suggestions and now just have to make some decisions. :). Again, thanks, all!!