what does he say to the orphanage staff?
What if they wanted to know more about the school like what subjects they teach.
He first tried to present it as a regular school transfer:
"No, I am a teacher," said Dumbledore. "I have come to offer Tom a place at my school."
"What school's this, then?"
"It is called Hogwarts," said Dumbledore
Then when it didn't work, he ObiWan-Kenobied them:
"Well, his name has been down for our school since birth —"
"Who registered him? His parents?"
There was no doubt that Mrs. Cole was an inconveniently sharp woman. Apparently Dumbledore thought so too, for Harry now saw him slip his wand out of the pocket of his velvet suit, at the same time picking up a piece of perfectly blank paper from Mrs. Cole's desktop.
"Here," said Dumbledore, waving his wand once as he passed her the piece of paper, "I think this will make everything clear."
Mrs. Cole's eyes slid out of focus and back again as she gazed intently at the blank paper for a moment.
So, he did in no way explain about wizards and magic.
Not that Mrs Cole needed much persuasion - she WANTED Riddle gone:
"He's definitely got a place at your school, you say?"
"Definitely," said Dumbledore.
"And nothing I say can change that?"
"Nothing," said Dumbledore.
"You'll be taking him away, whatever?"
"Whatever," repeated Dumbledore gravely.
how does Tom Riddle get to Diagon Alley? It is perfect reasonable that Dumbledore would have told him how to get there. Or there is a explanation that another teacher came to help him buy his things, but then again what do they say to the staff?
Riddle specifically didn't want any help, though Dumbledore planned to guide him:
"Where do you buy spellbooks?" interrupted Riddle, who had taken the heavy money bag without thanking Dumbledore, and was now examining a fat gold Galleon,
"In Diagon Alley," said Dumbledore. "I have your list of books and school equipment with me. I can help you find everything —"
"You're coming with me?" asked Riddle, looking up.
"Certainly, if you —"
"I don't need you," said Riddle. "I'm used to doing things for myself, I go round London on my own all the time. How do you get to this Diagon Alley — sir?" he added, catching Dumbledore's eye.
Harry thought that Dumbledore would insist upon accompanying Riddle, but once again he was surprised. Dumbledore handed Riddle the envelope containing his list of equipment, and after telling Riddle exactly how to get to the Leaky Cauldron from the orphanage, he said, "You will be able to see it, although Muggles around you — non-magical people, that is — will not. Ask for Tom the barman — easy enough to remember, as he shares your name —"
(HP and the Half Blood Prince, Chapter 13: The Secret Riddle)