At the camp where you begin the quest select your starting gear and check the supply box that is nearby to pick up some free potions and occasionally a selection of mission specific items. Then select a meal to give yourself the stat bonuses you'd like.
At this point you are ready to head out of camp and deep into the location to start tracking a monster. The first thing you need to do is search for monster tracks so you can use your scoutflies as a guide. Scoutflies are the special glowing flies that are part of your equipment that can identify tracks and markings and lead you towards your hunting targets. When you eventually find some tracks get close and investigate them. Doing this will slightly replenish the the bar on the right and allow you to locate your quarry.
It is important to remember that a monster must be defeated within a certain time so as you track your prey try to avoid any other monsters you may encounter as this will waste valuable time. As you follow the tracks collect any herbs and materials you find that can be used to craft valuable items. Eventually when you have discovered enough tracks the Scoutflies will lead you to the monster.
If at any point you lose the monster you are tracking you just need to open your map, and move the icon to the bottom left to tag it. It is also possible to select non-monster items on the map to also get a guide to their locations.