In order to boost Kiryu's Stats during the Main Story and thus become a better Fighter, you will need what is known as EXP, or experience. EXP can be earned in a variety of different ways, but the primary way of obtaining it is simply by defeating your enemies, especially the Bosses found within the game. You will also earn EXP by completing both Main Missions and Side Quests, as well as by playing Mini Games, eating Food, and by working out at the Gym.
Basically, you will earn EXP for every little thing you do in Yakuza 6; however, the amount of EXP you wiill earn will vary from one activity to the next. For example, you will earn way more EXP from defeating a Boss in a battle or by completing a Main Story Mission than, instead, playing a Mini Game or completing a Side Quest or even defeating a regular foe in battle.
On top of all of this, EXP will also be earned and separated into five different categories (as you can see at anytime in your Stats Menu). Red and Blue EXP will be earned from basic Combat with enemies, while Yellow, Green and Purple EXP will be obtained from completing the Story Mode Mission and Side Quests. You can even use vending machines found around both the cities of Kamurocho and Onomichi to buy drinks that will temporarily boost the EXP you will gain across all categories.