Fallout Shelter

After getting 18 dwellers in your Vault, you will unlock the Overseer's Office. This room will allow you to send up to 3 dwellers on Quests.

Each quest has requirements for the dwellers that can be sent, there will be Level requirements and either Weapon Damage Stats (the green crosshairs with a number, 5+ for example), or SPECIAL stats. There will also be a time, this is the amount of travel time it will take to reach the Quest area. It will take half of this time to return.

Assemble a team and equip them with proper weapons, outfits and pets as you have them, load them up with Stimpacks and Rad-Aways and send them on their way. I recommend sending your squad with at least 5 Stimpacks and 2-4 Rad-Aways, upping the amount they get as Quest level requirements get higher. Don't worry too much about Stimpacks as you will almost surely find a large amount while scavenging during your quests to stay healthy.

After sending your group out into the wasteland, it will take time for them to reach their destination. For some Quests, this time can be considerable, though this can, of course, be remedied for some of your Nuka Cola Quantum if you don't want to wait. If you choose to wait, you can log out and when you return you can select your squad and assume control over the mission.

The questing itself is pretty self-explanatory: You move from room to room, killing and looting any shiny object or glowing corpse until you reach the conclusion of your quest. Most quests will come to an end before you are done fully exploring the areas. It is recommended that you continue exploring, killing and looting as much as you can so you can bring back as many items and Caps as possible for your effort.

Occasionally, you will come across an enemy with a red skull. These are boss enemies, and they have more health and damage potential than regular enemies. Killing them usually results in some extra goodies, like loot or Quantum. There is a cumulative achievement tied to them as well, so it's in your best interests to go after them when you can:

As soon as you have 18 or more dwellers in your vault you will unlock the Overseer's Office, you will use it to send a person or group to a quest, where at the end of it you will have some rewards. You must complete 10 of them to unlock this achievement, I recommend you to send 3 people together at the beginning of the game, probably your dwellers should not be well trained yet.