Some of the most popular villages are Chipping Campden, Stow-On-The-Wold and Bourton-On-The-Water.
One of my favorite books I have on the Cotswolds is by Insight Guides; Great Breaks, The Cotswolds.
A small lightweight book that can easily be taken with you on your trip.$9.99.
Barnes and Noble has it, or you could order online.
https://www.insightguides.com/shop/product/insight-guides-great-breaks-cotswolds/9781780052236
It details many of the villages and mentions lots of things to do.
We did not rent a car last time we toured the Cotswolds.
We went with this company; Go Cotswolds:
https://www.gocotswolds.co.uk/
Guided day tours of the Cotswolds from Stratford-upon-Avon and Moreton-in-Marsh on Wednesdays, Thursdays, Fridays and Sundays. Excellent reviews on Trip Advisor:
https://www.tripadvisor.com/Attraction_Review-g186399-d6696369-Reviews-Go_Cotswolds-Stratford_upon_Avon_Warwickshire_England.html
There is train service from London to Stratford-Upon-Avon and to Moreton-in-Marsh, both are pick up points for the tour. The tour is in a Mercedes-type minivan and there were only a dozen people on our tour total. A small group.
We stayed in Stratford-Upon-Avon, but that was just our personal choice and I'm not pushing that on anyone else.
You asked:
"How much time do you all recommend I spend?"
It's hard to make a recommendation because I don't know what else you're trying to do on your trip.
It depends upon how much time you want to spend there; totally up to you.
Do you want to cover the Cotswolds thoroughly, going to dozens of villages? You'll need a car and 3 to 5 days.
Or do you want to take a one-day tour to get a sample of the Cotswolds, and will not have a car? Then a tour such as Go Cotswolds might be for you.
As Claudia said, the very popular Bourton-On-The-Water is very crowded in the summer.
Stow-On-The-Wold and Bourton-On-The-Water are located pretty much in the center of the map that James posted above, meaning either would be a good central location in which to base yourself if you will be driving a car around for 3 to 5 days to see other villages.