Chartwell in Kent is the former house and garden of war-time Prime Minister Sir Winston Churchill. It's now run by the National Trust. It has wonderful views and a really great vegetable garden.
Find out about Amorphophallus titanum aka the Titan Arum or the "Corpse Flower" - it's one of the largest, rarest, and smelliest flowers in the world.
Munstead Wood used to be the home of the great garden designer Gertrude Jekyll. It's also home to the garden she designed for herself.
Bodnant Garden in North Wales is one of the most beautiful gardens in the UK and is world-famous.
Great Dixter in East Sussex is an extremely popular plantsman's garden created by the late Christopher Lloyd, the gardener and author. I highly recommend it.