The 24-year-old Republic of Ireland midfielder completed his £9.5million move last week and flies to Russia on Wednesday to start his new life.
McGeady is keen to get away from the vitriol that he regularly encountered in his native city.
"Moscow is a huge city and it will be possible to have a degree of privacy," he told the Daily Record. "Glasgow really is a goldfish bowl and it's one of the things I'm glad to be getting away from.
"Neil Lennon (Celtic manager) said to me I might miss the adulation Celtic players get but I'm not so sure.
"My view is I need a break from it. I've had my experiences of Glasgow and I do need a change.
"There is always someone wanting to have a go at you. I got into a few fights in the past and wish I hadn't. I let my guard down a few times and it cost me.
"But some of the verbal abuse I had to take was really out of order. Because I'm a footballer, people think they are entitled to say what they like and get away with it.
"I'm talking about grown men here. They would shout, 'I hate you' or 'I'm gonna kill you, wee man'.
"There was also a lot worse but I'm not going to go into that."
McGeady claimed his decision to play for the Republic of Ireland rather than Scotland also made him a target for abuse from away supporters.
He told The Herald: "There are a lot of horrible places in Scotland for that type of thing: Tynecastle, Ibrox obviously is always going to be bad with the Celtic-Rangers rivalry, Motherwell, Falkirk.
"Some fans there hate everything Celtic stand for and everything I stand for as an Irish Catholic playing for Celtic.
"It begins in the warm-ups before games with all sorts of stuff being shouted at you, even from little kids."