Gallery of modern art in Royal Exchange Square is lovely, as is Kelvingrove in west end (right next to the fab park :)).
Shops - lots of lovely shops on Byres Road and surrounding area. Lots of nice places to eat too.
Down off Argyle Street in Finnieston there is a fab wee place called the Hidden Lane Tea Room which does great afternoon teas :) Saint Jude's on Barh Street also does fab afternoon teas!
For drinks I like Ashton Lane off Byres Road - Vodka Wodka, Radio/Nude, the Loft. There's a lovely cinema called the Grosvenor, has a bar attached so you can take drinks in, also does sofas you can use instead of seats - bit more pricey though. There's also Ketchup next door which does loads of amazing burgers!
Loads of Italian type places to eat everywhere really.
Again down Argyle St at Finnieston lots of lovely wee bars - was in The Ivy last week and the bartender knocked up a fab cocktail made specially to my liking, was yum!
I like to drink at Slouch on Bath St - indie music and amazing cocktails yum yum.
Ohh just thought of Transport museum - just been built, lots of cool bits ad pieces, will be overrun with kids though.
Lots of beautiful Charles Rennie Mackintosh buildings e.g. The Art School and a church on Maryhill Road - not sure when they're open though.
Princes Square in town has lovely shops indoors, more expensive end of things IYSWIM and places to eat/drink e.g. November on top floor.
For shopping there's Buchanan Street or further out Braehead which is near Renfrew, you can get the ferry down the Clyde from town though!