From London, what would be the best way to get to the Cotswolds on public tranportation? Also how do you get around them, with public transport?
Take the train from Paddington station in London to Moreton-in-Marsh, it's an hour and half ride. While in the Cotswolds you can use the bus, but I have to warn you that there isn't any bus service to the smaller quaint villages. You won't have any trouble getting to Bourton-on-the-Water, Broadway, Stow-on-the-Wold and a few of the larger towns. For the Cotswolds renting a car is really the best way to see the Cotswolds.
Richard, You might like to take a walking tour of this area. Many agencies, for example, www.ramblers.com can provide guided or self-guided tours. They arrange hotels, bagage transfers, maps, etc. I spent a week by myself doing this, and it was a wonderful way to see the Cotswolds.