Is there any the preference:
Establish climb / descend then initiate turn
Or turn, then climb / descend?
If both are ok, which should be learned first?
Cheers!
Both are fine, and at the end of your PPL training you should be able to demonstrate the ability to do all of it - turn then up or down, or up or down then turn. Once you get out of training and into regular flying with ATC, you may be asked to do any or all of it (e.g., "climb to 9,000 and turn left heading 270" or "turn heading 270, descend 9000", or even "slow to 190 knots, then descend 3,000'. Through 5,000', turn right heading 300.")
As for which should be learned first, it doesn't really matter. They're all core concepts that you need to learn, and it doesn't really matter which order they come in.
With that said, this is the program I followed when teaching as it seemed to have worked well for the decades that came before me:
The first lesson was always straight and level as this is the most fundamental exercise.
Then, we'd move into learning climbs and descents as that's how we get up to and down from straight and level flight.
From there, we'd teach turns so you can learn how to turn from one straight and level path to another.
After that, we teach the student how to combine them - climbing turns and descending turns. For the PPL I always focused on entering the climb or descent first, then starting the turn as this was often more comfortable in terms of g-loading for the student. For the CPL and IFR world, however, the student must learn the forces on the body by turning and climbing/descending at the same time, so I would expect them to do both at the same time. Not saying I never had a PPL student do this, but it was not out of the gate.
The next exercise was into the circuit where you use all the exercises described above. The circuit is where we perfect hand flying skills as it all happens quick, and you have to be able to move from a climb to a climbing turn to straight and level to a descending turn to a decent all in the space of about 5 miles. It's easy at the end, but in the beginning trying to stay within +/- 20' and +/- 5 degrees is quite hard.
Understand that flight training is all about the progression from the basics to advanced concepts and the reason ab-initio pilots get into trouble is that they try and get too advanced too quickly. At the end of a PPL, a climbing turn seems really basic, but when one breaks it down, there is
A LOT of physics going on with the airplane, so the core components must be mastered first.