A list of places you need to find and the activities you need to carry out to complete the Bastion Exploration Challenges: All ten Bastion Regions Challenges or Bastion Collectibles Challenges such as Trailblazer, Hidden Messages, Footsteps of Idris, Vassa's Triumph, Mederines' Disciple, Mederines' Order, Mederines' Peer, Grave Historian, etc.
In total you have to find 103 locations: 63 Districts + 30 Landmarks + 10 Hidden Places (there are more of them but only ten of them are needed to complete all the challenges).
When you find all these places, you'll complete the Exploration Master Challenge.
You need to find 80 arcanist runes if you want to complete all arcanist runes challenges. You'll find most of them during outdoor freeplay. A few more you will find by visiting hidden places a.k.a. dungeons.
You need to find 80 misplaced writings if you want to complete all misplaced writings challenges. You can find all of them during outdoor freeplay.
The good news is that the items needed to complete these challenges can be farmed. This means that you can collect the same items again, for example after you die during a freeplay session. The bad news is that not all the items you need will appear in the same places in every freeplay game. Additionally, they can be collected by another player before you do so.
Outlaw, Dominion or Scar items can be found in Outlaw / Dominion / Scar related Camps that usually can spawn in areas that Outlaw, Dominion or Scar enemies can spawn in.
Arcanist and Sentinel Archives can both be found in Camps that will usually spawn outside of "normal" faction spawning areas.
A few examples (not all) of how the places where you can find the necessary items can look like.