A producer can easily become overwhelmed by the amount of issues to deal with. And each issue is important to the person posing it. So a producer needs to figure out a system for addressing everyone's problems, yet at the same time, prioritizing them so they aren't quickly suffocated by them. This system develops from real work experience, figuring out what works and what doesn't through addressing problems on each production and consulting with mentors, and building a team around you that you trust to help with any problems that occur.
In general, you, as a producer, need to be concerned with the big picture. Let your production leaders do the lion's share of worrying about the smaller picture. You should consult with each department regularly and get overall assessments on how the department is doing. If all seems to be going well then continue to trust that you have hired a competent department head who can handle any issues that develop within his or her team. Micro-managing is death to you and your production. And, believe me, any unresolved or festering problems will make their way to your door eventually.
When you are consulted and I promise you will be on a daily, even minute to minute basis, you need to figure out a way to provide solutions quickly and effectively and calmly. The people coming to you for guidance are already stressed out and anxious. They need you to be the strong, calm one in the crisis. At the same time, they need to be reassured that they have the means to handle the problem themselves. If they feel confident in their choices, and much of that confidence comes from a producer's validation and trust, then they will perform at the highest level for you and need to bog you down less with issues. Being calm doesn't mean you have all the answers all the time, it just means you are confident you will be able to find those answers that allow you to keep the project moving forward despite any obstacles that come your way.