You can overwrite the error message in the __init__ method of your serializer:
self.fields['field_name'].error_messages['error_message_key'] = your_custom_error
You can find all error_messages keys related to each Field in the documentation. For instance, CharFields error message keys are required, max_length, min_length.
If you're using custom field from DRF:
name = serializers.CharField(
...,
error_messages={error_message_key: your_custom_error}
)
If you want to raise more specific errors, I encourage you to take a look at Raising ValidationError from the official documentation and ValidationError from DRF documentation.