What put the three of them on to their search was a meeting with a junk shop owner who Mundungus supplied. The locket, Voldemort's Horcrux, had passed through his hands. The man's description fit the boys to a tee. Hermione surmised, and Harry and Ron agreed, that Voldemort must have used them to find his Horcrux after he had found that it was tampered with, and before Draco and Blaise were known criminals. The threesome set off at once, tracking them across the continent and into the Mediterranean.
In Italy they found their first solid piece of evidence: a cheap motel that the two had stayed at. They had met with a man who might have been Snape, or MacNair, or any number of Death-Eaters. They had left early that morning, and sent their payment back in cash. They had left no forwarding address.
Harry checked into the room and scoured it for clues while Ron and Hermione tried various shops in the area. The boys must have stayed indoors during the day and been careful not to speak to anyone too closely at night; several of the bartenders thought they might have served them, sometimes with another man, sometimes without. But nothing certain. They left disappointed, ready to backtrack and try another route.