Public Transport from Oxford to Cotswolds, Stratford-Upon-Avon

According to rom2rio.com you can take a train from Oxford and get off at the second stop Leamington Spa and walk over to Leamington Spa Railway Station to catch a bus to Stratford-Upon-Avon. The total trip takes 2h. Stagecoach Midlands is the name of the bus service.
What towns interest you in the Cotswolds?

I looked into this as I'm visiting family in England this Christmas, and was hoping to tack on some time in London and Moreton-in-Marsh: Moreton is a good base for that side of the Cotswolds. From Moreton there is a quick train to Oxford, a bus to Stratford, bus one way to Broadway and Chipping Campden, bus another way to Bourton and Stow (and a walk between the two Slaughters), train and walk/taxi to Blenheim Palace and Woodstock - as I remember, there may also be a shuttle to Blenheim from the closest train station.

Bus to Blockley and Hidcote Manor. Kingham has a couple of very good but expensive pubs, an easy walk to Daylesford Farms. I stayed in Kingham many years ago.

Note, from what I can tell, bus schedules change: more frequent during the summer months.

If you stay in Oxford, you'll probably use a couple of days for Oxford itself. Blenheim and Woodstock are closer. The villages mentioned above will probably take a couple of hours to reach.

Stratford-u-Avon is about an hour and a half from Oxford by train - change at Leamington Spa. Service is fairly infrequent though, two-hourly.

You can also get off the Leamington Spa to Stratford uopn Avon trains at Wilmcote to visit Mary Arden's Farm
The journey planner also shows the option of changing at Leamington and Solihull for Stratford
For the train to Moreton in Marsh and buses to places such as Chipping Campden you should have a look at the Cotswold Discoverer Ticket

You can take the train early morning from Oxford to Moreton-In-Marsh and be picked up by this tour, which I highly recommend. This is much easier than trying to catch a bus from one town to another (then another bus to a new town & repeat) in the Cotswolds.
https://www.gocotswolds.co.uk/product/cotswolds-in-a-day/
They give an excellent tour of the Cotswolds.
Check days and times on their website and make reservations.

We have traveled with Go Cotswolds and were pleased with our day tour.
You could possibly go to Moreton early enough that you'd have time to walk around before being picked up by the tour.
They will drop you back in Morton about 4:30-5:00 in the afternoon, so you might have time to again walk around and see some of the town before taking the train back to Oxford.

We have traveled from Oxford to Moreton by train.
We've also traveled from Oxford to Stratford-Upon-Avon by train; had to change trains at Leamington Spa.

We spent a couple of nights in Stratford-Upon-Avon, as we wanted more than a day trip there.
Not far from S-U-A are Warwick (the town) and Warwick Castle, both worth a look.

The tour suggested by Rebecca is the way I saw part of the Cotswolds. It allows you to cover quite a few small towns without being at the mercy of the bus schedules. Bus routes are typically set up to get locals from their small towns into the larger town (Moreton-in-Marsh in this case, either as a destination or as the place to pick up the train to Oxford), not really to connect small towns to each other unless they happen to be nearby and along the same road. It's a serious challenge for a tourist to get to more than two small towns in one day, which is a shame when the towns are really tiny and can be visited rather comprehensively in less than two hours.

GoCotswolds uses vans, not big buses.