The setting initially is in London. Sherlock Holmes and Dr. Watson live on Baker Street in London. Helen Stoner traveled into London to see them. The actual crime was in Surry, England. The county of Surry is southeast of London. She says that,
"I started from home before six, reached Leatherhead at twenty past, and came in by the first train to Waterloo." (pg 3)
The town of Leatherhead is about twenty miles from Baker Street, which is the famous address of Sherlock Holmes. She says that she lives with her stepfather who is
"....the last survivor of one of the oldest Saxon families in England, the Roylotts of Stoke Moran, on the western border of Surry." (pg 3)
There is no place in England named Stock Moran, but there is a place called Stoke D'Abernon which is a few miles from Leatherhead.
The page numbers I have given you are from my internet copy of "The Adventures of the Speckled Band". However, you should find these quotes in the first fifteen paragraphs of the story.
The story begins at the house on Baker Street in London at a time when Sherlock Holmes and Dr. Watson are still sharing a flat. They meet with Helen Stoner on the ground floor because it would be improper for a young woman to come to their rooms upstairs. Helen tells them a long story about the death of her sister Julia. Shortly after Helen leaves, her stepfather barges into their flat and threatens Holmes with violence if he doesn't stay out of his affairs.
The setting then changes to a decaying manor called Stoke Moran. It is located only a short distance from London. Dr. Roylott, who has a terrible reputation for hostility and violence in the neighborhood, lives there with his stepdaughter Helen. His other stepdaughter Julia died under mysterious circumstances two years earlier. The building is in a state of decay and only a small part of it is occupied. The doctor, who spent many years in India, collects exotic animals and allows a cheetah and a baboon to run free on the estate. He also allows a band of gypsies to camp on the few acres that remain of the once large estate.
Holmes and Watson rent a room at a nearby inn where they can see Stoke Moran. That night, after receiving a signal from Helen Stoner, they cross over to the manor and climb through Helen's bedroom window. She goes off to sleep in a different room while they wait in the dark to see what will happen.
Most of the important action takes place in the room being temporarily occupied by Helen and in the adjacent room occupied by her stepfather. These are both described in detail. Of importance is the fact that Helen's bed is fastened to the floor so that it cannot be moved, that there is a ventilator between her room and her stepfather's, and that there is a dummy bell rope leading from the ventilator down to the bed. In Dr. Roylott's room Holmes and Watson see a steel safe, a saucer containing some milk, and a dog leash with a noose tied at the end.
Stoke Moran is a cold, sinister place. Dr. Roylott is half-insane. The atmosphere is oppressive in a gothic way. The building and the grounds are neglected, mainly because Roylott is having serious financial problems and cannot afford to make all the necessary repairs.
The story ends back at 221B Baker Street, where Sherlock Holmes explains to Watson how he deduced that the doctor had murdered Julia Stoner with a poisonous snake and was trying to murder Helen Stoner in the same way and for the same motive. Julia had planned to be married, and Roylott would have had to give up her one-third share of the capital he was controlling under the terms of his deceased wife's will. When Helen Stoner became engaged two years later, she began hearing the strange whistling sounds that Julia had told her about just before she died in agony, talking incomprehensibly about a "speckled band." The ventilator was an important clue, as was the useless bell rope. Holmes deduced that the snake was kept in the steel safe and the doctor used the whistle to call it back through the ventilator if it hadn't yet bitten his intended victim.