Depending on your interests, you might consider:
Greenwich -- A short boat ride to the naval museum, observatory, Cutty Sark, and by now the Painted Hall restoration should be complete in the naval college.
Windsor -- A short train ride with a castle that combines antiquity and current-use splendor, plus the beautiful St. George's Chapel (where Harry and Meghan married last year) with many royal tombs. Plus nearby Eton if you want to see that.
Hampton Court -- A short train ride, longer boat option, home of Henry VIII and the later Stuarts, with distinct architectural styles.
Canterbury -- Train ride to one of England's most beautiful and historic cathedrals, in the medieval center of town.
Salisbury -- Ditto but the cathedral is outside the center in a meadow-like setting, also with an interesting museum. Stonehenge is nearby with frequent bus service.
Oxford or Cambridge (not both unless you really like universities) -- Each a train ride with beautiful architecture and fine museums.
Do they have to be day trips? Can you leave your bag with the hotel, pack a few overnight things, and spend a night elsewhere? If so, Canterbury, Salisbury, Oxford, or Cambridge would be more enjoyable with more time to explore. Bath would be another option with Roman ruins, a beautiful abbey in the center of town, and lots of Georgian architecture. So would Winchester, another cathedral town with lots of history, and Portsmouth, with a naval museum and historic ships.
It's hard to "explore the countryside" without a car. There are lots of local buses to small villages but you have to follow schedules and some are infrequent, limiting your flexibility.
