Once you have at least 18 dwellers in your vault, you can build the Overseer's Office for 1000 caps and begin to send dwellers on quests.
Some quests are part of a chain and must be done in order to unlock more quests. It is possible to encounter special quests during standard wasteland exploration.
Upgrading the Overseer's office allows you to attempt multiple quests at once.
Level 2: 3,500 caps (requires 30 dwellers) - Send dwellers out on 2 quests at a time
Level 3: 15,000 caps (requires 55 dwellers) - Send dwellers out on 3 quests at a time;
To unlock the achievement complete 60 of these quests, which does take quite a while as it takes real-time hours for dwellers to travel to and from the quest destinations.
Hope this helps!