World Traveler Achievement in Little Kitty, Big City

To start this quest, you need to talk to Tanuki near the construction site. He'll require three feathers to get started. These are from catching any three birds for Bird Botherer (25G). The game seems to imply that some sort of bread source is required to catch birds, such as getting humans to drop sandwiches or bagels, but that isn't required. Just sneak up to them from behind them with  and then press  to pounce when the prompt is on screen. Get three feathers and talk to Tanuki to start this quest, opening up your first portal. For all other portals on the map, each one only requires a single feather. Keep a stock of 3-5 feathers and you'll easily complete this quest while exploring and working on everything else. The portals themselves aren't marked on your normal map for some reason. You can only see them on your fast travel map when you enter a portal. Here is a video showing where they all are though. The one on the far east of the map is the one you'll likely be missing, since it's not intuitive to get to.