As has been said, Didsbury, Chorlton. Didsbury still popular with medics generally.
if you’re looking to pay less then some of the old ‘student’ areas – Whalley Range, Withington (‘north Didsbury’ as some would have it) close to Chorlton and Didsbury.
there’s Northenden too. Close to Wythenshawe. Not too far from Didsbury and Chorlton.
Places like Gatley and Stockport would also be close enough, possibly cheaper than Didsbury and Chorlton. Not as pretty. Not as easy for town and some of the bars and restaurants that might appeal.
trail riding is in short supply. A few woody bits with some small bits around and about. But there is the Mersey to ride alongside. Marple, Poynton, Macclesfield, and thus the edge of the Peak are close enough to ride to if you have time.
House shares are very common. Check the local estate agents or use rightmove . And check some of the specialists in shared houses – often targeted at students but also catering for young professionals.
For example https://www.homes2share.co.uk
good luck
edit edit ‘green spaces’. Plenty of those in south manchester.