Day trip in October

Bath, Canterbury, Brighton, Oxford, Cambridge, York if you're willing to suffer a long day.

Unless it is a part of the RS tour, consider a day trip to Bath by train. One you get there the main parts of the town are very walk able. Get an early start and return late as there is plenty to fill up a day there.

No need for a tour company either (unless you really want one). You can easily get to some great locations by train on your own. Windsor, Bath, York, Canterbury, Dover, Winchester, Stratford-upon-Avon are just the start. What are your interests? There are also some great things to do in London that might not be included in the RS tour. Greenwich, Kew Gardens, interesting gardens and cemeteries.... and lots more. It will help us make relevant recommendations (and maybe give you specifics about trains, buses or tour companies) if you tell us a bit more.

We enjoy the knowledge shared by taking a tour. Would love to see more castles, countryside, etc.
We would prefer leaving London since we will be there 7 days with RS and have free time to explore other London sites. Thanks for all suggestions.

There are free walking tours in Bath, starting in front of the Roman baths. I don't remember the schedule, but thats all we needed. You can walk to there from the bus or rail station easily enough. Most of what you have time to see on a day trip is pretty central anyway, although no castles or countryside in town.

London Walks does day trips to various worthwhile places outside London -- see the "Great Escapes" link on their website. You don't need a reservation, show up at the designated station, buy your "day return" ticket (a bargain), and pay the guide 18 pounds (16 if you're >65). I guess you pay for lunch too. I haven't done one of these but I have done some of their in-London walks and they have a well-deserved good reputation.

Get lost exploring the colleges of Cambridge or Oxford, something definitely best done without a guide. Take a train from London. Easy.