Congratulations! This part isn't easy.
I notified my chair first and 1-2 people I was very close to in the department. I then immediately told the dean/provost as well (VP of academic affairs / dean of faculty — same thing as provost at many schools). Then I told the rest of my department. Then I told friends in other departments. I did all of this in person except for the dean, which was over the phone (because she was super busy), and I made a very strong effort to have all of these meetings happen within days / one week. I really went out of my way to make sure I saw all of these people in person and spoke to them as rapidly as possible. And being at a small school I knew the rumor mill would travel quickly so I wanted people to hear it from me, not someone else.
My advice is to tell your department chair first and then go out from there in order of importance in terms of relationship to you (plus dean/admin being early on in the telling, possibly the second person.) And if your department is smallish (10 or less), then try to hit up your whole department in person, and before you branch out much to friends in other departments. I also recommend as many in-person tellings as possible. If you know you're not going to reach certain people in person then you can e-mail or call but do try to make the effort. One person I had to tell in my dept was on sabbatical and was one of those cut-myself-off-from-the-world people. I had to go REALLY far out of my way to figure out a number to call her but I'm glad I did and she was glad I did as well.
I guess I'm saying just try your best to leave on as positive terms as possible. Congratulations again!