I'm not local, I'm Italian like you, but I just came back from Scotland. I visited Endinburgh, Glasgow and Inverness. What I can tell you is that Edinburgh is a wonderful city and definitely a "must-see" if you go up there. In Edinburgh I liked to discover the city walking in the center, you should go to the Castle, walk in the Royal Mile and also discover Holyrood Palace. Everything is wonderful.
Secondly, I don't suggest you to visit Glasgow, I didn't like it at all, even if I was prepared to see an "average" city. Too modern and "gray", no particular places that struck me. I've been there few hours, so maybe I can't say much about Glasgow.
Inverness was the real surprise of my trip. Small and lovely town, full of particular and traditional buildings, I stayed in an amazing B&B in Fairfield Road, a street all dedicated to B&B. If you go there, you should visit the path along the river with the "Islands": the famous Ness Islands Walk. Something that make you think that it is really worth to move there. Also, Loch Ness is there, few miles away. I suggest you to go by bus and visit "Urquhart Castle".
I was also lucky, because the weather was perfect and I enjoyed my stay.