How to properly inject services into other services
Alright, I think I figured it out. It should look something like this: 

I couldn't find the service name anywhere for MenuTreeParameters so it's omitted in the examples for now. 

I setup my menu service in rest_api.services.yml

services:
  rest_api.menu_service:
    class: Drupal\rest_api\Services\MenuService
    arguments: ['@path.alias_manager', '@menu.link_tree']


And created my service: 

namespace Drupal\rest_api\Services;

use \Drupal\Core\Menu\MenuLinkTree;
use \Drupal\Core\Path\AliasManager;
use \Drupal\Core\Menu\MenuTreeParameters;

class MenuService {

  protected $aliasManager;

  public function __construct(AliasManager $aliasManager, MenuLinkTree $menuTree) {
    $this-&gt;aliasManager = $aliasManager;
    $this-&gt;menuTree = $menuTree;
    $this-&gt;items = array();
  }