I think 3 days is perfect for the Cotswolds, especially if you like hiking/biking. If you base yourself in Stow-on-the-Wold or Moreton-in-Marsh you will have lots of public transportion options. There are several bus lines that run throughout the Cotswolds-I've never needed a car.
You could spend a day seeing Stratford upon Avon and see a play at the Royal Shakespeare Theatre (if you like Shakes.) or visit Warwick Castle which is nearby. Trains run from London to Moreton-in-Marsh and National Express runs buses to Cheltenham Spa for as little as £4 each way.
York is another great mini-break. Make sure you visit Castle Howard - it's beautiful. If you book far enough in advance you can get train fares from London for £10.