Drupal - How to Reduce/Increase module's weight on the module install process?

Use hook_module_implements_alter() rather than changing the module weight.

Sample implementation from content_translation.module:

function content_translation_module_implements_alter(&$implementations, $hook) {
  switch ($hook) {

    // Move our hook_entity_type_alter() implementation to the end of the list.
    case 'entity_type_alter':
      $group = $implementations['content_translation'];
      unset($implementations['content_translation']);
      $implementations['content_translation'] = $group;
      break;

    // Move our hook_entity_bundle_info_alter() implementation to the top of the
    // list, so that any other hook implementation can rely on bundles being
    // correctly marked as translatable.
    case 'entity_bundle_info_alter':
      $group = $implementations['content_translation'];
      $implementations = [
        'content_translation' => $group,
      ] + $implementations;
      break;
  }
}

There's an API for this now:

module_set_weight('your_module_name', 10);

You can also implement the hook as Ivan Jaros said, which allows for more fine-grained control (e.g. first for one hook, last for another, after a specific module for the third). But the module weight should work too.

If you use import/export configs, you can change module's weight in core.extension.yml file, number after module's name is weight.