Pollokshields, Shawlands, Strathbungo, Newlands are all possibilities on the south side of Glasgow. They're mostly leafy suburbs (especially West Pollokshields, some parts of Shawlands, and most of Newlands and Strathbungo).
I'm in West Pollokshields and my neighbour is a consultant who commutes to Crosshouse (not in a specialty that usually requires urgent on-call). When we lived in Shawlands, I commuted both to the south side of Ayr and (for a later job) to Kilmarnock. But as a PP has said, it's easier in the other direction.
Have a look at conversions: I'm not sure about the rental market but you might be able to find something in your budget with a garden/access to garden. We're in an 3/4 bed, 2 reception rooms, dining kitchen upper conversion (half of a large Victorian villa) and have a lovely large garden (back garden split left and right, front garden technically communal). I'll not tell you how much we paid for it 22 years ago as you'd cry 
A friend bought a similar, but slightly smaller house along the road, with the garden split slightly differently (he has the back half of the back garden plus a strip up the side for access, plus a small area at the front): 2 reception rooms, 2 bedrooms, kitchen & bathroom. About 6 years ago it cost him, iirc, £300 or £330k.
Looking further ahead for your family, Glendale Primary is very good. Shawlands Academy is also very good - but not all of Pollokshields is in its catchment. We had to do a placing request for ds to go there. At the time most/all placing requests were accepted but I think that's changed now. (Ds is now at Aberdeen Uni 
).