This is an interesting area of Glasgow. You will definitively have to visit the oldest builings in town, which are Glasgow Cathedral and Provand's Lordship house. Also a good walk around the Necropolis is not a bad idea.
Below is information on a trip around Historic Glasgow from everyone's favorite treasurer, Neil.
Historic Glasgow Tour: Glasgow Cathedral, The Provand’s Lordship House, St Mungo’s Museum and The Necropolis
Come and discover Glasgow’s impressive history with a visit to the city’s oldest buildings. Admission to all locations is free of charge, although there are generally donation boxes at each entrance. Donations are at your discretion although I think I’ll probably donate a couple of pounds. We are going to meet at 1 pm outside the main entrance of the Cathedral. However, I’ll stand outside the entrance to the City Chambers in George Square at 12.30 pm in case anybody isn’t comfortable finding their own way there. It should just be a 10 minute walk to the Cathedral. I’ll try to remember to wear my International t-shirt so that you can recognize me. Directions direct to the cathedral can be found here:
If for any reason this link doesn’t work then you can go to www.mapquest.co.uk. Enter Glasgow in the city search box and G4 0QZ in the post code box. For some reason, if you search for Glasgow Cathedral directly it gives you slightly incorrect directions.
We hope to get a guided tour of the Cathedral and then we shall visit the other sites. I estimate that the entire tour will take between 2 and 3 hours but you can come and go as you please. We’ll end our cultural experience in the usual GUIS way by going for a couple of drinks somewhere nearby.