The question what staff members you need, and how good they have to be, depends partly on your strategy. But there are a few staff members that you will need in any case:
You always need a coach for your team.
He gives your players an advantage during the matches.
He also improves the tactic training of your team.
A physical therapist
The physical therapist improves your players' energy regeneration. Without a physical therapist your players will quickly get low on energy, leading to injuries and deficits for their training.
An Assistant Coach
Training your team is one of the most important aspects of the game as it brings your team ahead. The assistant coach gives and advantage to your players' skill training and should be hired as soon as it is possible. (He needs the second level training facility to be hired.)
The better your staff members' skills, the more they help your team. But good staff members can get pretty expensive, so don't lose track of your spending.
Over time you will need other staff members as well, but some of them only become available later on after you construct the required facilities.