You can send out your dwellers to the wasteland to explore. Simply drag a dweller outside the vault entrance and you'll have the option to change his/her equipment and increase his/her stimpack and radaway supply. When ready, tap the "Explore" button to send him/her off.
While exploring, they'll earn experience in encounters, find Outfits, Weapons, and caps. The longer they explore, the better gear they'll find. To ensure their survival, you have to consider the following:
Explorers will use Stimpacks automatically. You can check their progress from time to time. Make sure to recall them when their Stimpacks are running low. The time they'll take to come back is half the time they spent exploring.
If ever an explorer dies in the wasteland, you can spend caps to revive him/her. The amount of caps depends on the explorer's level. The maximum amount is 1000 CAPS for reviving higher level explorers. There's no "expiration" for dead explorers so if you don't have enough CAPS to revive them, just leave them there and revive them when you have enough.