Training StaffThere are two types of staff that affects training speed: Trainer and Youth Trainer
TrainerYou will have this dude by default, although his level will be very low at the start. In order to train players, you should get a higher level trainer, at least Superior. Here is a list which shows the difference in effectiveness of trainers of different levels
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Optimally, it would be good to get a world-renown (level 7) trainer. However, they can be very expensive (up to 3 million) and their upkeep can be quite expensive. So many people go with at least level 6 (exceptional) trainers if they are training for the NT (or if they want to cap out a MVP/HOF).
Remember to watch out for the staff's salary when bidding on him. You might be able to get a world-renown trainer for $1000, but if his weekly salary is $100k, then you might find yourself bankrupt soon.
Youth TrainerOnly get them if you are training players aged 18 or 19. They won't work on players aged 20 and above. They're much cheaper than trainers, so if you're serious in training players with MVP or higher potential, you should at least get a level 6 youth trainer. Again, watch your finances. You don't want to go bankrupt just because you can't wait to train your next franchise player. The difference in speed of Youth Trainers is still pretty unknown at the moment, so there isn't much data.