One thing is working on PMs skills.
The trick which helps PM to learn delegation is trust. If I think why we prefer to do the task by ourselves it's usually because we believe we'd do it better, which basically means we don't trust that someone would do it at least equally well.
However it also means PM doesn't give people chances to develop their skills. As long as they aren't given challenging tasks how can they earn PM's trust? It's a kind of vicious circle. So PM basically needs to break it, making step forward and assuming people will cope with tasks they're given. Not only is it a chance to build trust and develop them but also it a good lesson about delegation for PM.
As long as PM is good with delegation the rest comes pretty easy as they should have enough time to work on communication, coordination, task management, etc.
Another thing is (not) having a good example.
One of reasons we often suck when becoming PMs or people managers is we don't have good role models around. Especially when we don't bring a long experience with us, we often learn from people who we're working with.
So if my manager sucks at delegation chances are good I'll suck at that as well. If PMs around micromanage their projects and focus on individual tasks instead of coordinating effort of the whole project team I'm likely to do the same.
What PM can do here is to find a good role model. Look around, search for PM who act like you'd like to act. Ask them to mentor you. It's a very light-weight relation, yet the one which should be very rewarding for both sides. In terms of personal development a good mentor in the workplace can hardly be overrated.