Start of the quest is by talking to Lucas Simms or Mister Burke
FOR DISARMING THE BOMB
The required skill to disarm the bomb is Explosives 25, for a quick clean disarm without using Mentats, use the level up skill points to boost your 'Explosives' level to 25 after leaving the Vault, than head straight for Megaton and get to work. Mentats are back up, but the stuff and questes in Megaton will get you leveled up quite quickly
Lucas Simms offers 100 caps for disarming it, 500 by speech check, or Good Karma for disarming it for free. And apart from payment/karma he gives you a key to your Megaton house.
Before disarming the bomb, you can get a good Karma bonus for locating and killing Mr. Burke, Killing him in Megaton won't make others hostile to you and you get access to his belongings, including a Silenced 10mm pistol. Do NOT tell Lucas about Burke, for Lucas will get shot by him.
Keeping Megaton on the map and having good karma, the citizens give you free items, like aid, food, ammo and caps once a day
BLOWING UP MEGATON
*P.S.A: This option can not be done in the Japanese version*
You need the Fusion Pulse Charge and that can be acquired by Mister Burke and a Expolsives level of 25 (Boost by Mentats). After equipment the device, Burke will meet you at the top of Tenpenny Tower. There he and Tenpenny will give you the honour to detonate the bomb and turn Megaton into a big hole in the ground. After which, he gives you 1000 caps and a key to a Tenpenny Suite... Also a major karma drop
BLOWING UP SIDENOTE
Before detonating the bomb, it is recommended to go through Megaton and collect useful loot like books, aid, ammo and weapons. One being very important, the Strength Bobblehead inside Lucas Simms house, Also a Karma drop bonus for killing the citizen's.
Quest-wise: Make sure you know the location of James from Moriarty, for the 'Following in his footsteps' quest. If you have started or about to start the Wasteland survival guide, Moria Brown, the quest giver, survives the explosion, but becomes a ghoul and relocates to Underworld.
A possible companion: Jericho survives unharmed
One drawback to the quest is either the Talon Merc.'s (Disarming) or the Regulators (Detonating) hunting you down