As a new faculty member, how do I avoid the Peter Principle?
Based on my experience with faculty, if you care about managing well, and don't want to be a bad manager, you are already doing better than many people. Again, in my experience, the worst offenders along these lines are those who think they are God's gift to their graduate students and postdocs, and cannot bear to hear any criticism. 

So, if you want to do a good job managing: 


Keep caring about doing a good job and encourage constructive
criticism/suggestions.
In particular, talk to the people you are managing, and find out
what they think. Lots of junior people are scared to even give
constructive advice to their seniors, because they don't know if it
will adversely affect them. It is obvious to say that you can often
learn a great deal from the people around you, but people sometimes
forget that this is very much true of the junior people you work
with as well.


This is only tangentially relevant, but this blog post by Matt Zimmerman is quite good.