1- Kelvingrove Art Gallery & Museum
Kelvingrove is the most visited museum in Scotland and the most visited in the UK outside of London. It recently underwent a massive refurbishment which has added new collections to its already impressive invitory.
2- Museum Of Transport
Everything from a horse and carriage to vintage steam trains are available here. A great place for kids or for a family day out, plenty to see.
3- Glasgow Cathedral
Worthwhile just to see the building itself, it is hundreds of years old and still looks magnificent, not bad inside either.
4- Burrell Collection
Located opposite the Kelvingrove Museum and well worth doing along with its more popular neighbor. Plenty to see from Ancient Egypt to information on Sir William Burrell who donated the collection.
Great selection of tips, although the Burrell Collection is actually on the opposite side of Glasgow from Kelvingrove. It's easily reached by train or bus from the centre of town, though. The collection is housed in a purpose built museum in the middle of Pollok Park. The Park also contains Pollok Country House, which is worth a wander, and a herd of Highland Cattle, which is worth a photograph or two.