This page of the Desperados 3 guide explains a potential problem - why doesn't the game show any challenges to complete? This can be problematic if you haven't play Shadow Tactics, Mimimi studio's previous title. If you are familiar with that game, then you should know why the challenges (Badges) don't appear initially.
Challenges are hidden actions and minor tasks that you can do while playing a mission. Each completed challenge rewards you with a Badge - the game informs you when you get one (example in the picture). You need to complete all challenges if you want to have 100% completion of a given level.
The first mission of Desperados III is a prologue and, therefore, it doesn't have any challenges. It has only the main objectives.
The reason why challenges related to a given mission are invisible is very simple - the game reveals challenges only after you have completed a given mission for the first time. Then you can find out how many of them you managed to get, and how much more is left to do.
A very important information is that you can complete challenges during your first playthrough of a given mission. Our guide has world atlas for all missions - read them in advance to learn about the challenges. You can complete them even when the game hasn't revealed these challenges yet.
Unfortunately, it is very unlikely that you will complete all challenges in a single run. Of course, you can complete the remaining challenges by replaying a given mission. Lowering the difficulty level (preferably to Beginner) can also help in unlocking some challenges, especially if the challenge involves, for example, falling within a time limit (speedrun) or walking through an area without attacking anyone.
Repeating a mission also helps in tracking your progress in completing a particular challenge/badge - they are found in the last tab in the journal. The example above shows that there are 2 failed challenges. Unless a given badge is about completing a mission without saving progress, you can always try to load one of the previous saves and get that challenge.