I haven't lived in Glasgow for a while so I may be a tad out of date.
Rogano in the city centre is worth a try, there's a fine 1930s style bar and restaurant upstairs with a less fomal but very good "cafe" downstairs.
The Ubiquitous Chip in the west end is a good place for Scottish food, in season they have a good selection of game. There's another similar restaurant nearby but its name escapes me, I'll add it if I remember.
Stravaigin (three or four in the city) is a god pub / restaurant chain. I believe that it was founded by a former chef from The Chip.
The Hotel du Vin in the west end is also worth a look, it used to be One Devonshire Gardens and the last time I ate there Gordon Ramsay was calling the shots in the kitchen.
There are a number of good Indian restaurants, restaurants that offer authentic food rather than spicy junk for drunks at pub closing time.
The Three Judges (at the bottom Byres Road where it meets Dumbarton Road) is a good Glasgow boozer, it has a super selection of guest ales and they are always in good condition.
The Station Bar in the city centre (near the Theatre Royal) is another good basic boozer, it gets busy when performances at the theatre finish.
Shopping - I don't know what sort of shopping you want but, despite being a sometime resident of Glasgow, the place is a disaster (living in Tokyo has tainted my perceptions). Having said that, there's..., er, um. Gieves have a place in Edinburgh.
The underground railway is one of the oldest in the world, it's tiny and has one loop.
The Burrell collection on the South side is worth a visit.
Be aware that the city centre can get rowdy at night, many of the city centre pubs are unpleasant in the evening.
Even if you aren't an art nerd (Lord knows I'm not!), you have to check out the Museum of Modern Art. Small enough to handle in an afternoon or less and if I recall, pretty close to the center of town.