I also was in your boat, I was promoted very young and was looking for ways to gain trust and respect from my crew. I took all the classes, read all the books, but nothing is better than experience. It will take some time, but like Scott said everyday let your crew see that you can make a decision, tough or easy. No one wants an officer that can not make a decision. Eventually you will get some type of call that you will need to perform a task with your crew or make a critical decision. Only then will they see that you can do the job. One call will turn into two and so one. When they see your performance on the emergency scene and that you can make a decision, and will not get them hurt, then you will eventually gain their trust. It takes time and remember lead by example.
Also some good books on leadership are Chief John Salka's "First in Last Out” and one of my favorites is by a Navy Captain D. Michael Abrashoff “It's Your Ship: Management Techniques from the Best Damn Ship in the Navy”
Good Luck

David,
As a young officer, the first bit of advice I can give you is to know who is around you who you can take "suggestions" from. As an officer, when time is of the essence, you have to make the call, and the men have to follow. They should, afterall, they put you there. However, more often than not, there is at least a few seconds to ponder what needs to be done. There's nothing wrong with seeking input from older and more experienced members in times like these.
As far as books to read, my suggestion is to always be reading one. I think the best book to start with as a company officer is John Norman's Fire Officers Handbook of Tactics. Also a good read is Responding to Routine Emergencies by Frank Montagna. I feel that those two books are a great place to start because they have the nuts and bolts of firefighting and cover most situations you will find yourself in. Beyond that, start looking into the strategy and incident management oriented books. Fireground Strategies by Anthony Avillo, Fireground Size-Up by Michael Terpak, and Fire Command by Alan Brunacinni are a few I recommend. The list goes on and includes Fireground Safety by Vincent Dunn and Random Thoughts by Tom Brennan. Many more.
Keep reading, keep training, and keep talking with members of your department and the fire service in general. It's important to constantly keep your head in the game, especially if you aren't seeing a lot of fire in your department.

Stay Safe,

Jay

David, Before being promoted I wanted to know what my boss expected of me. I believe that the best thing a new officer can do is to set expectations for his or her crew. A new officer has many challenges to face from the new position, I found that by informing my crew about my expectations I wasn't in a position of playing catch up when a crewmember did something I didnt care for. I found that any of the books listed above where worthwhile to read. Congrats on your new Job