You could certainly get out to any one of the home counties and back in a day using public transport.
If you only have a day the best option might be to join a day tour because instead of wasting time with connections and time tables they will take you there and back so you can see more. I've never joined one but I assume you get herded around like a mob of sheep and with an annoying guide it could just be painful not fun.
These are some of the day tours from a company called Golden line
Bath, Stonehenge, Salisbury and Magna Carta
Windsor, Bath and Stonehenge
Oxford, Stratford and Warwick Castle
Leeds Castle, Canterbury & White Cliffs of Dover

I don't know if any of these would be what you're into but it's just a sample. There are hundreds of tour companies based in London and one of them is sure to have an itinerary that appeals. Even if you prefer to go it alone having a look at the itineraries of these companies might give you some inspiration.
Usually the Lonely Planet guide books have sections with suggested day tours etc. Maybe borrow one from your local library.