Have a look at Pollokshields (Nithsdale Road mentioned above is in Pollokshields). The first planned suburb with lots of lovely big stone built villas, big wide streets, large gardens, trees and parks.
Newlands, the other side of Queens Park (and yes, that is a park
) also has nice Victorian era houses.
Shawlands - in particular "High Shawlands" - also has some nice houses. Strathbungo is a conservation area with some lovely townhouses.
Once you've lived in one, with their high ceilings and large rooms you get spoiled for "modern" houses.
For that budget, you might not get a full stone villa, but many of them have been converted. We live in one - dh calls it a "horizontal semi" and have 3 bedrooms, a study and a big dining room (which could be a bedroom) as well as a large dining kitchen and an enormous living room. The garden (even just our half but we choose not to have a boundary with our downstairs neighbour) is of a size that in London would have been sold and 2 4 bedroom houses squeezed on to it)
Shawlands, Newlands and most of West Pollokshields feed into Shawlands Academy, which is a good school. We actually had to do a placing request for ds for Shawlands Academy as we're in the catchment for Glendale Primary (which has an excellent reputation but you might struggle to get in in a in-year request), as it feeds to a different secondary but to date, all placing requests have been able to be requested.
On that note - catchments are key in Scotland. Every child is effectively in the catchment of two schools - a non-denominational school and a Catholic school. (There's also a Jewish primary in Newton Mearns). However, it is possible to do placing requests to a different school which may or may not be accepted, dependent upon space. We thought we'd have to do a placing request for Glendale but it turned out it has a strange shape of catchment and we were already in catchment.
Southside has good links into Glasgow (and in Pollokshields, our wee diddy subway) and lots of lovely parks - Maxwell Park, Queens Park and Pollok Park.