To get Beast Tamer achievement/trophy in Final Fantasy 13-2, you need to befriend Don Tonberry, a rare monster that can be found in Bresha Ruins AF100, west of the area where Atlas stood in Bresha Ruis AF005.
He will appear as ‘Void’ before getting engaged. If you can’t find it, turn ‘Battle Mania’ fragment Skill On to increase his spawn rate.
You can unlock ‘Battle Mania’ via Chocobo Girl in Serendipity after getting ‘Paradox Doctorate’ fragment from the City of Academia AF400, which requires 100% map completion for all areas in the game.
If that doesn’t cut it, turn on ‘Monster Collector’ fragment skill to increase the odds of befriending monsters. You can unlock ‘Monster Collector’ via Chocobo Girl in Serendipity after you have earned all fragments from Vile Peaks AF010 & AF200.
Fight Don Tonberry on easy difficulty to increase the odds of successfully taming the monster. If you fail to befriend him the first time, wait for him to re-spawn and battle him again until you have recruited him. It will take few tries even with above two helpful fragment skills.