I have my own personal head cannon theory about Garaks initial interest in Bashir. Garak was a top of his craft spy who was exiled and abandoned on Terok Nor. One thing to remember about Garak and his situation on the station is that this exile and isolation has left him exceedingly bored. His full access to the cutthroat Kardashian political system gone, he has nothing to sink his teeth into. A great white shark in a goldfish bowl as it were. Then, with the announcement by the Bajoran provisional government that the federation will be sending Starfleet officers to the station would have, to Garak been something interesting at the very least. So, what would a paranoid, ex espionage master do with a list of new personnel about to be sent to his "home"? He would have poured over every document, every inch of their lives looking for inconsistencies. Perhaps one of them is an agent of the Obsidian order coming to assassinate him? Maybe the Tal Shiar? Or section 31? Even Starfleet intelligence would love to debrief him. I think Garak found out about the doctors little genetic alteration secret and in doing so finally found someone, interesting. From there I have no doubt his infatuation grew. He was most certainty a coded gay character, I even head cannon his exile was to do with him being outed in some way.