Qu. for training school
1. Prices - make sure it includes all the landing fees in the rates (would be expensive otherwise when doing circuits).
2. Same aircraft: not only same type but actually the same aircraft or just a few. I had 10 different PA28s when I did my PPL - had to adapt every lesson to different radios, some with sprung controls, diffferent Vx/Vy, some with ASI with MPH/Kts others with Kts only, different avionics [esp. adds to the work when doing some radio nav], some avionics 'known not great on this aircraft' [e.g. AH, DI drift] but 'good' on others. Not a problem now but not what you want when learning.
3. Instructor availability - key (or being able to book in far enough in advance, with extra lessons because you will always lose a fair proportion to weather).
However, I would say the most important thing is to have a good instructor who you get on with, who you can stick with. (How you find their teaching you can only know by flying with them at least once/a few times.)