As far as the Board goes, the director's role is to provide the trustees with the information they need to make informed decisions. This includes that sense of excitement about the future that the director can bring to the Board's deliberations. This isn't about rejecting what the institution has done in the past; rather, it is using the institutional history as a springboard for the future. The director and the trustees need to be simpatico, having a shared vision of the future. They also need to have clearly delineated roles, although from everything I read about corporate and nonprofit governance, the line between the board's job and the director's job is changing fast. There was an article in the Wall Street Journal last Monday (January 14) about this. The thrust was that CEOs (and, I think, library directors) will go further with their boards if they level with them about major problems and ask their advice before acting.
In a political sense, a public library director's job is not to make trouble for the elected officials of his or her community. Don't do anything that would reflect badly on the elected officials, and if something bad may be about to happen, give them a head's up early. Also, get every elected official involved in every GOOD thing the library does: "READ" posters with the mayor's picture, ribbon cuttings with the entire city council, offering the library's meeting room to state and US senators and representatives who don't have funds for a local office...all of these ingratiate you to the power structure.
During your interviews with trustees and staff members, I'd ask the staff and the board, "What do YOU expect from the director?" The greatest statement you can make if you want to be known as a good conversationalist is, "Tell me about yourself." Give them a chance to talk about it. Then relate what they tell you to the better bosses you've had, or make a point of showing what you've learned from bad bosses. (Not mentioning any "bad boss" names or locations, of course...no matter how tempting it may be.)