Agreed with nbt and admin’s suggestions - the canal is the total off-road route but possibly quite a bit slower and circuitous (and puddled in places).
Coming up towards Rose Hill on the route admin posted, you can emerge on Stockport Road and immediately use the right-hand turn lane towards Middlewood Way (or the pavement and crossing), which makes it surprisingly unintimidating.
I’d then definitely recommend using Woodville Dr to climb all the way up to Hibbert Lane - quieter than Buxton Lane and somehow feels less of a climb! If you wanted to avoid traffic again, you could even then take an immediate left off Hibbert Lane (avoiding the speeding cars swerving to avoid the awful humps) and climb the hill round the back way to emerge right at the foot of Goyt Mill. Small diversions round back roads like this can really help make cycling more appealing.