Ohhhh fun!
Prepare for a wall of text!
Well, like woadgrrl said, Kelvingrove and the Science Centre are good places to see. The Hunterian is also fantastic.
The Hunterian Art Gallery is small but usually has a nice selection up, plus the Mackintosh House is attached to it and is pretty great. The Hunterian Museum with all the weirdo odds and ends (William Hunter was an 18th Century Anatomist
) is across the road in the Glasgow Uni main building. Most of it is shut to the public right now because they're repairing the roof but they have the must-see stuff out in the main entry hall. The Uni looks like Hogwarts, so that in itself is worth having a glimpse at.
Unfortunately the Transport Museum is shut because they're moving it to a new location which will be the Riverside Museum. They have the Harry Potter car there and I love it.
There's also GOMA which, to be honest, can be hit or miss. If you don't even like modern art, don't bother.
There's the Burrell Collection and Pollock House which are out in a really nice, leafy part of Glasgow. They've got a group of heeland coos out there too and I think there are some calves born in the spring if you need a dose of cute.
You HAVE to go to the Botanic Gardens because they have a whole room in one of the greenhouses devoted to carnivorous plants and it's awesome. I also used to live just near the gardens so I am unapologetically biased.
Other than the touristy stuff, there are lots of nice pubs and restaurants in the City Centre as well as in the West End.
I don't know, I could probably go on for ages. If you want anything more specific, feel free to PM me!