How do you learn the craft of showing character emotions in a subtle manner? For example, in movies like The Tokyo Story, the character emotions and reactions are very subtle. So how do you show character emotions in an effective but subtle way?
Via their actions and their words. Show what they're doing, rather than telling us flat-out how they're feeling.
Watch people. Watch their eyes and their micro expressions, watch the way their move their hands, the way they move their shoulders and shift their weight. Failing that, read about body language and learn from it. It's simplistic but something like this might be a good place to start. You always need to think about how specific characters emote - many things are just human nature when it comes to nonverbal communication but someone with a louder personality might have more dramatic expressions. One person might visibly furrow their brow, clench their teeth, and glare to show anger while another might just tense their shoulders and look away, and there are plenty of other ways to express anger both loudly and quietly. But yeah. Watch people, mostly, and really pay attention.
I do not use terms like "angry" or "sad", since I instead use body language. Lots of little visual clues and actions which make up the expressive whole person. Body language is often left out of writing, but the manner in which something is presented means more than the words themselves. I can tell you a character is sad or I can show it.