If you want to see London, recommend staying within walking distance of the key site like Parliament, the Tower, etc.
If you want to visit places outside London, but stay there, there are a few places like Cambridge, Canterbury, Windsor Castle, Oxford, etc. where you could take the train or a bus to those places and still stay in London.
Going farther afield, like Bath, the Cotswolds, Salisbury, Westminster, Stratford Upon Avon, Warwick, etc. I recommend away from the city. We used Chipping Campden in the Cotswolds as our base to visit Stratford Upon Avon, Oxford and Blenheim Palace, as well as the Cotswolds. Recommend the Volunteer Inn. It has the best pub in town and a great Indian restaurant.
Bath is another place you might visit and stay. We visited Wells and Glastonbury from there. The Brooks Guesthouse is nice and provides parking. Don't miss the fish and chips at The Scallop Shell. Best fish and chips in England.