Welcome to the city of culture!
For a cultural tour I would suggest:
1. The Kelvingrove Art Gallery andMuseum for paintings, (they have the Salvador Dali painting "Christ of St John on the Cross" which even as an unbeliever, is truly stunning in the flesh; the "Glasgow Boys" and many outstanding others) plus lots of other museum type stuff - Spitfire, Dinosaurs! (some stuff loaned from the Burrel Collection whilst the latter is undergoing refurb).
2. The Hunterian Art Gallery and Museum (right in city centre) house collections of, amongst other things, CR Mackintosh work. (see The Macintosh House).
3. Gallery of Modern Art (what it says on the tin).
4. The Peoples Palace - collection of objects, photos etc of how Glaswegians lived, worked and played. (Billy Connolly's big Banana Boots included!)
4. The Riverside Museum is worth a visit (used to be the Transport Museum) but has been subject to modern practice of arranging things for people with ADHD. i.e. there is no theme such that you could follow through the development of the railway locomotive or ship building industry in Glasgow. Stuff is just presented out of any context alongside other random stuff. The "Glenlee" ( a large barque) is good.
Take tea in the Willow Tea Rooms in city centre (aka Miss Cranstons tea room) - all things CR Macintosh.
Some good pubs and restaurants in the Merchant City - too many to list.
If you are after other recommends just ask