A coach's tendencies are not always observable at first. Sometimes, it may take multiple games to discern a head coach's style of play, other times it takes single drive.
Some coaches run their teams like a business. If you aren't getting results, you can expect to sit on the bench. Good effort is rewarded and poor effort is met with extra practices and discipline. The classic example of this style of coach is Jimmie Johnson, famous for champships at the NFL level with the Dallas Cowboys and at the collegiate level with the Miami Hurricanes.
Other coaches take a more player oriented approach. They work with the talent they have and try to develop it into a winning team. Players often prefer to work in this environment.
Just as important as the head coach is the offensive and defensive coordinators. The coordinators are both the engineers and drivers of the offense or defense their team has. They design the team's playbooks. It's their job to get with the head coach and make sure the system works. Often they have a preference for a certain style of offense or to utilize certain players. They must be flexible in designing a style of play that suits their expertise with the talent resources available. They also are responsible for calling plays in game.