If you want flexibility, you're going to need either a car, or a decent road bike. The Cotswolds are very nice for cycling, so as long as the weather holds don't discount it!
In terms of public transport, there are two railway lines running through the Cotswolds. Along those, moving about is fairly easy. Trains are fairly frequent (especially by rural line standards), fares are pretty good (especially with railcards like the Cotswolds Line Railcard). The two lines are Hereford / Malvern / Worcester / Evesham / Kingham / Charlbury / Oxford (northern and eastern Cotswolds) and Swindon / Kemble / Stroud / Cheltenham (southern/western).
For buses, there are quite a few, but perhaps not enough. There are a few longer distance bus routes crossing the Cotswolds, such as Oxford to Cheltenham. There tend to be routes linking the places with stations to those without, such as the Cotswolds Railbus. The larger towns have bus routes serving the surrounding villages, such as those heading out from Witney or Cheltenham.
The problem you might find is with crossing between areas of fairly good public transport. That and Sundays... For example, if you were based in Witney, you can fairly easily get out to Burford, or to Oxford station for the train, or up to Charlbury to take the train to the Wychwoods. Getting over to Cheltenham ought to be fine. However, from starting in one of the surrounding villages, wanting to visit those over in the west, expect to have to "hub" through places like Oxford, Witney, Cheltenham and Swindon, and don't plan on there being any sensible connections or joined up timetables available.
All the trains take bikes (reservations needed one a few peak trains only). So, give some thought to getting bikes, making your way along one railway line doing excursions to interesting places, then zip along to another edge and head back up the other way!