What is hard to earn but easy to lose? Trust. That is a famous riddle, probably. Anyway, Trust is an integral part of Days Gone because it allows you to gain access to new items in the various camps around the world. Each camp has its own Trustometer (unofficial name) that lets you know just how far they think they could throw you.
You can gain Trust in a few different ways. The largest chunks of Trust come from completing one of the many, many, many missions or bounties that Days Gone offers you. Finishing them off will garner you some Trust for whichever camp the mission happens to be helping out, so this is a good way of getting started.
Once you reach a point where the missions are no longer for a particular camp, you have to look elsewhere in order to gain more Trust. You can grind out Trust in a couple of different ways but both of them involve a bit of grinding and some item collection.
Perhaps the most efficient way to gain trust is to go hunting for wolves. If you take Wolf Meat and hand it in to the kitchens within the camps you can boost your Trust with that camp. For example, if you give them to the kitchen in Copeland's Camp you can gain 15 Trust per piece given in. Plus, you get some credits every time you do this too.
If you would prefer not to reduce the population of wolves to near extinction - and you'd best believe that should never be the aim - then you can hunt Swarmers instead. Swarmers are a basic enemy in Days Gone, if you defeat them and loot them you can hand in their ears in order to gain a little bit of trust. It isn't much per ear, but the enemy is so common that it offsets that downside.