I believe Part-time training is only when the club is semi pro or amateur, so just check the general info tab and that should tell you. In this case, you should turn professional when you reach league 2
As for training, train them in the roles you are going to use them as, so if your tactic has an AP, train the players you are going to use there as an AP and so on. Young players just train in the role you want them to be, but key attributes for most positions are teamwork, decisions and technique. Make sure your staff are as high level as possible also.