First Visit to Glasgow-- Where should I start?

Suggest that you stay in the West End which is near the Botanics Gardens and Byres Road. It fist all your criteria and is just a 1 minutes underground ride from the City Centre...

Right that should be 10 minutes from the city centre

Places to visit
Botanic Gardens
Kelvingrove Museum
Transport Museum
Glasgow School of Art
Art for an Art Lover
Burrel Collection
Glasgow Cathedral
Museum of Modern Art

We stayed at this flat in the West End earlier this year and found it excellent value.
http://www.unique-cottages.co.uk/cottages/south/glasgow/the_park_apartment
Depending on when you are going it may be within your budget. It is certainly bigger than you need but it is in a great location and is very well equipped. No internet, but there is free wifi, and good coffee, nearby at the Offshore café.

Another place in the West End, that we have not used, but which might suit you is:
http://www.queensgateapartments.com/home.php?id=1

alihutch's list is a good one, my version would be similar but probably wouldn't include the Botanic Gardens, and would add the Provand's Lordship house, the St Mungo Museum of Religious Art, Glasgow University & the Hunterian Museum, and the Tenement House. The Burrell is a highlight in my opinion and is 5-6 miles out of town, as is the House for an Art Lover - they could both be seen in the same trip. The City Chambers, on George Square, is worth a visit if you can be there when they have the free guided tours (Monday-Friday I think). Maybe take a stroll through the Merchant City. Good coffee at Tinderbox - in Byres Road and in town - and try The Wee Curry House in Ashton Lane if you like indian food.

All good suggestions from GregY2....
And even better coffee and Eggs Benedict at Heart Buchanan (which is a fab deli) at the top of Byres Road! Actually Burrel Collection is only 3 miles from the city centre and you can get there by a very short train ride from Central Station....just feels further out as it's surrounded by so much greenery.
If you like designer clothes then Merchant City is your spot, as well as lots of renovated buildings ...this website
http://www.glasgowmerchantcity.net/trail.htm
tells you all about the area.

actually I lie about the Burrel Collection distance...about 5 miles!...but still a quick train ride....

thanks for all the good information-- a great starting place. i really enjoyed the merchantcity information-- especially the history page. both apartments are a bit big for me. i like the idea of having a kitchen-- but really only need a very small room in a good location. thanks again fodorites --
kawh sebastopol, ca

I would add the Hunterian Museum and the Hunterian Art Gallery, the latter for its Charles Rennie Mackintosh house. Since only part of the house is open at the moment, admission is free. Note that the Hunterian is closed on Sundays.
I would skip the Hop On = Hop Off bus. It's pricey as transportation and you don't really get close enough to see the things the narrator is describing.
Our view of Glasgow was somewhat distorted because DH developed tendonitis which limited his walking considerably. But if you have the misfortune to need it, I highly recommend the medical attention at the Western Infirmiry.