Agreed with the advice on training but I strongly disagree on avoiding C's and lower.
For one thing you cannot actually avoid them as there are not enough Bs and As to fill up your farm system. Secondly Cs can be contributors. Also if they get hot their potential can increase.
For the OP's question, progression is determined by 4 things:
potential, age, performance, and training. Even if you train Joe Baseball in contact, his other attributes will increase as well. Only his contact ratings will increase at a faster rate.
Playing time does matter, so if you got a 19yr old A potential guy rated low try to avoid putting him in A baseball. In your example I would have one in either AA or if possible move one to SS. It is more accurate to say performance matters instead of playing time though.
Best advice I could give is to periodically check up on your prospects. If a guy with promise struggles heavily and for an extended period of time in AAA consider dropping him to AA. Also try to avoid calling guys up if they would primarily be on the bench. Yes, a 74 overall SS is MLB ready, but if you have a 80 overall above him leave the kid in AAA.