I was always a Amon fan in 3, so... all side stories. But seriously, keep in mind doing any side story gives you good XP, and XP is what levels you up and unlocks better moves (XP works differently than 0 and Kiwami 2) so if you want the combat at its best, make sure you do as many as you can.
But stuff you wanna keep an eye out for:
Chapter 4/5 has a couple of side stories that focus around Dragon Palace - these start the quest chain to get all of Komaki's moves. Do so before chapter 5's boss to dramatically improve the combat. The quests should start when you visit the dragon palace and clear out the goons blocking the entrance.
Murder at Cafe Alps - this one starts, surprisingly enough outside of Cafe Alps. Speak to the protester. Basically if you want a mini Phoenix Wright game in your Yakuza, do this.
Tuna Fund - forget how this one triggers, but it spans both cities and multiple chapters and has some decent rewards. I just really enjoyed it for some reason.
Perfect Cup of Ramen - this one starts at Kyushu No.1 Star and again spans several chapters and locations. Basically the story of one man, a stubborn chef, and the quest for the perfect store brand Ramen.
For the trans positive side story, make sure to visit the massage parlour a few times, then Bantam.
I think there's a plot that starts in Shellac that has call backs to the first chapter, first battle of Yakuza 1. Its a really cool side story and well worth tracking down.
All the mystery of Ryuuku side stories in Okinawa are absolutely worth doing. They are usually triggered by a group of school girls.
There's one that focuses around Date, I think it triggers in New Serena? Its actually pretty important to his backstory going forward, do track it down if you can.
If you want more wacky Akimoto adventures, then there's not one but two to see. One in Okinawa I think is impossible to not trigger, and one in Kamurocho later in the game. You need Haruka for both.
Mikiyo gets two side stories that build up his backstory. One fairly early on where you patrol the market with him, and one that focuses on his upbringing. I forget how to trigger them though.
Rikiya also gets a duo of side stories (in Kamurocho). Do both of them, you need to have him with you, so that should narrow down the exact points in the game you can do them.
Well, I'm tired and wrote all that from memory, so I refer to those reasons as to any inconsistencies or omissions in my recommendations.
