What about taking a train to Cheltenham Spa? That is a lovely town to start your journey. Lots of shops, restaurants, theatre and fine buildings. From there you can get a bus to Bourton on the Water, one of the prettiest towns in the Cotswolds. Then, taking the same bus, move on to Stow on the Wold and after that to Moreton in Marsh. Depending on how much time you spend in each of these places, you could go on to Stratford-upon-Avon. One of my favourites, but then I used to live there! All these places will be quite crowded in high season though.
The bus from Cheltenham runs two hourly but you might have the check that. You could just go as far as Moreton in Marsh and take a train back down to Reading and further South.
I've done this without a car quite easily, You do need to book hotels/airb&b well in advance. If you are without a car, make sure you know how far your accommodation is from the bus stop/railway station. Travel very light!
Allege rape in Epsom by asylum seekers


