Parties that are not in use can be sent on an adventure by themselves. However, they will be unusable in normal battles until either their adventure is complete or you destroy the party.
Go to your "Camp>Parties" screen and make sure you have an extra party created. When you do, click the adventuring image above the creatures' names to get started.
Either break a party or click the adventuring image to cancel an adventure.
All adventures have a 22 hour wait. So you won't be able to find out until later how well your team did!
After your party's adventure timer has expired, just click the adventuring image to complete its adventure and get your reward (if any). You can read about how adventuring rewards are determined right here.
Provided you don't already have a different party adventuring at a given dungeon, you can press this button to repeat a party's previous adventure.
Adventuring parties have the following limitations: