Unlocks in this game follow a similar style to older EG games, where there's a mix of obvious and more obscure methods for different characters. To unlock characters in this game generally, there are two confirmed and known methods that you should go for right now, plus some evolving ideas:

Each rank in the Challenge Mode part of the game has a match play boss, which will unlock that new character for World Tour mode. After that, you simply need to complete that character's first mission in World Tour to unlock them for play moving forward.

In addition, all of the caddies in this game are gated behind Challenge Mode, so if you're looking to get someone who isn't Hannah, you'll want to play through this ASAP to find your favorite. If you haven't found them, they are unlocked within a specific challenge in a few of the tiers, so complete ones you've passed to hunt them down (or look further down in the guide).

Some characters and World Tour missions appear to be locked off by needing a character to hit a certain Loyalty level (needs confirmation - myself and others report some as low as 4-6, others as high as 20 or 30 from what I've seen). If you've gotten stuck and have been playing as a single character for most of the game, you'll need to play many others and get them up to speed. Sometimes you'll get enough Loyalty through playing the World Tour mode, but otherwise you'll need to grind more rounds anywhere else in the game to get the Loyalty you need.

(Needs confirmation, currently speculation) Some characters appear to only be unlockable after completing certain chapters of other characters, in addition to other prerequisites. Try completing more chapters - in the worst case, you'll raise your Loyalty which will help in general.

She is available at the start of the game. :)
To unlock Hugh, become the champion of the Community Golf Open in the Beginner tier of Challenge Mode.
To unlock Rune, become the champion of the Amateur Tournament in the Amateur tier of Challenge Mode.
To unlock Lun, become the champion of the Mistral Cup in the Pro tier of Challenge Mode.
To unlock Arcana, become the champion of the Nature Park Cup in the Bronze tier of Challenge Mode.
To unlock Sebastian, become the champion of the King's Cup in the Platinum tier of Challenge Mode.
To unlock VCE-3000, (Speculation, taken from Wiki) beat Ekaterina in the Secret 3 tier of Challenge Mode.

Unlike the caddies, golfers have a mix of unlock methods to obtain them. Some are simpler, while others are more obscure and (at the time of writing) are still being figured out. (I'm currently digging through metadata and the public client code to find confirmations, but tracing assembly takes some time, sadly). As mentioned above, the best things to do for uncertain unlocks are to beat Challenge Mode tiers and complete World Tour missions, with playing the game and gaining loyalty across characters being the next best thing at the moment.

Starter character!

Starter character!

Complete ALL the tutorial missions so you get the achievement. Then, complete "My First Swing" on Alie's World Tour.

Beat Aya at the end of the Beginner tier of Challenge Mode, then complete her first chapter in World Tour.

Beat Lani at the end of the Novice tier of Challenge Mode, then complete her first chapter in World Tour.

Beat Sasha at the end of the Amateur tier of Challenge Mode, then complete her first chapter in World Tour.

Beat Sophie at the end of the Semi-Pro tier of Challenge Mode, then complete her first chapter in World Tour.

Beat Cougar at the end of the Pro tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Max at the end of the Bronze tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Garuda at the end of the Silver tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Dino at the end of the Gold tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Izzak at the end of the Platinum tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Gangol at the end of the Secret 1 tier of Challenge Mode, then complete his first chapter in World Tour.

Beat Lilith at the end of the Secret 2 tier of Challenge Mode, then complete her first chapter in World Tour.

Beat Ekaterina at the end of the Secret 3 tier of Challenge Mode, then complete her first chapter in World Tour.

(Needs confirmation) Increase Aile's Loyalty and play through their story missions.

(Needs confirmation) Increase Mizuki's Loyalty and play through their story missions.

(Needs confirmation) Increase Lani's Loyalty and play through their story missions.

(Needs confirmation) Increase Sasha's Loyalty and play through their story missions.

(More characters to come!)

(Spoilers below, there be dragons)

From personal experience, I unlocked the Secret 1 tier by finishing Cougar's World Tour missions. It's possible this is triggered by something else, though.