This is either one of the longest or shortest quests in the game, depending on how you wish to finish it. Speak with Moira in the supply shop at Megaton to acquire the quest. If you take the long route, you will have 10 various tasks (see the link below) to complete that will take you from one end of the world and back. If you choose the short option, you'll finish it in the same conversation as you start it. If you need the experience, go ahead and do the entire quest. If you don't, talk your way out of it.
To talk your way out of it you will need a high Speech skill. So either focus on that early or come back for this quest later in the game. Ask her about one of the missions and then ask why she can't do it herself. Eventually you should have an option to tell her the idea isn't good and she should not write the book. Once you convince her of this the quests ends and you will still get the achievement without running all about the Wasteland.
Warning! It is possible to lose this quest if you blow up Megaton during another side-quest. Moira will survive as a ghoul initially, but she may be killed by Raiders on her way to the Underworld once she turns into a ghoul. To be safe, disarm the bomb when doing "The Power of the Atom."
See The Wasteland Survival Guide for more specific info and all 10 missions.