Using the Expedition Center, you can send your Champions out on expeditions to find supplies - particularly equipment. Along with finding supplies while traversing the wasteland, the Champions can also find Secret Locations - special solo missions that often have very good rewards.
There is still a lot we don't know about how stats impact the survival of Champions and loot found during expeditions.
It may look like going from Expedition Center Level 1 to Level 2 only makes the expeditions harder, and you may be tempted to stay at level 1. Do not be deceived! Along with better loot, some Expedition Center upgrades include additional Expedition Slots.
Trust and power system was reworked at some point, will continue to update these numbers as I improve my shelter. Leaving the old numbers for now instead of a bunch of ?.
When starting out, expect to mostly do Short Expeditions. Your Champions will be too weak, and eLDi too limited, to manage longer Expeditions.
You can choose between doing a lot of Short Expeditions for more frequent guaranteed loot - but longer expeditions give better guaranteed loot and more chances to find Secret Locations with less micro-managing. Which is more important to you is for you to decide.
Send plenty of eLDi, especially if you intend to try exploring Secret Locations. Do not worry, as any unused eLDi will be returned upon completion of the expedition.
There are currently 11(+?) Secret Location maps, all of which can be quite deadly early on. If your Champion has close to the recommended power, expect to use a good bit of eLDi.
But if you have the time and strength, the Secret Locations are a good source of supplies, including the potential for a lot of Nanites.
There has been talk from some of the developers about adding up to 20(!) Secret Locations soon. Some of those may be easier than the existing maps, but many will likely be more dangerous.
Send Champions with a good equipment and a mix of stats since we don't know which stats matter.
Use Crate Champions for Expeditions, not Story Champions. Story Champions will sometimes be required for Campaign Missions, which will be blocked if you send them on an expedition. Crate Champions are never required for Campaign Missions, so the only thing blocked are their individual Trust Missions.