How to get your audience to care about your characters?
First you have to know the difference between Character and Characterization.
Characterization is everything that can be described about a given character: Appearance, Beliefs, Personality, Mindset, etc.
Character is the choices a character makes under pressure. (like the phrase, 'so-and-so has good Character', it refers to what a person does, not what a person can be described as.)
Normally characters act according to their personality/beliefs and Characterization when they're not under pressure or at risk. However when something THEY care about is at risk, or they have to risk something to get what they want, that's when they should reveal their Character.
And here's the answer to how to get people to care about the characters: When their Character and their Characterization are at odds, are different, they become interesting. They become "3-D". They reveal that there is more to them than what the audience can see on the outside because their choice seems to go against who they SEEM to be.
The stronger the contradiction between Character and Characterization, the more interesting they become and the more the audience will care about them.
This intrigue and interest is what makes readers care about characters. It'll make them want to know more, to see your character undergo trial after trial and reveal to them who the character truly is.
Because if a character only ever acts like you expect them to, that's what makes a flat character and thus a boring and predictable character. And Readers don't care about boring and predictable things. They want to be surprised so there is suspense and tension and the ever-present question of 'what are they going to do next?'
Thus, a general principle is that your Principle characters ought to have well-developed Characterizations and an accompanying, contradicting Character.
Simple Examples of Good Character:
- A seemingly kind and peaceful, well dressed character actually acts like the most ruthless of villains.
- A seemingly scary and serious character, once interacted with in a pleasant manner ends up being kind of dorky and shy.
- A character that steals things for fun and appears selfish, ends up giving their food and money to people who have less than them, thus ending up with less than they started with, showing a noble Character.
- A sweet character that starts off as a friend and does everything to support the protagonist--when the pressure is on, they end up only caring about themselves and stabbing the protagonist in the back.
Followup note: Character can have an arc through the story, and often they should. Things they do that match their Characterization early in a scene can start to change as they undergo conflict and pressure, thus revealing who they truly are. And through the arc of the story they can change who they are.
Example: An otherwise good man goes through increasingly trying circumstances and ends up the most ruthless of villains (Breaking Bad). OR A villain hell-bent on capturing the protagonist realizes the futility of his quest and decides to help the protagonist instead (Zuko from AtlA).