For this achievement, you’ll have to send your Monsties on Excursions at least 10 times.
To do this, you must first unlock the ability to send your Monsties on Excursions during the Canalta Timberland/Sheparden story arc. After using your Canyne to Ground Dive to a new area and defeating the Feral Monster at the end of it, return to Sheparden and speak to Master Kol to tell him you’ve completed your task.
After a bit of story development, you’ll regain control of your character outside your camp in Cataracts, and, upon entering your tent, you’ll have the Excursions option unlocked within the camp menu. Select it and send whichever Monstie you want out on a little adventure. They’ll return with experience and improved skills, so this is a great way to enhance your Monsties’ stats and give them Environment Skills, particularly Ratha, who has neither! Highlight the location to see which Environment Skills and State Increases will be learnt.
Unlike previous Monster Hunter games, where you send out a team of your Monsties (or Palicoes in the non-Stories games), Excursions work on a one-Monstie-at-a-time basis, and require the use of a Training Talisman. You get 3 to begin with, and can buy more from the Melynx Emporium’s Special Items Selection with Silver Trade Points.
Once you’ve sent 10 Monsties out on Excursions, you’ll have your achievement!
Like many of these achievements, you can check your progress in your Ranger Records by pausing the game, navigating to “Ranger Files” (one tab to the right), and then selecting the Ranger Records dossier (the red one on the right). The Excursion Training Manual medal is the sixth medal on the fifth row.