Did you triggered a cutscene inside the Guild's building (the one indicated by a Wheel on the map)? If not, you'd need to complete some more tutorial quests (the ones that are told by the butterfly). Once you've got them done, just talk to the butterfly in your home's room, accept it and you'll advance the story a bit. If you trigger an intro video while sleeping, you should be able to use the gate afterward.
To access Internet Play, go in the Guild's building, walk all the way up until you see three guys behind the counter. The left is for StreetPass, middle for Link features (this can activate Link messaging, kinda like how BF feature in ACNL operates), and right for Multiplayer. Talk to the latter guy, select "マルチプレイ", then "インターネット". If you want to seek "open gates", pick the first option (and hit the blue button if there isn't any yet). Pick the second option if you want to "open your gates". Be sure you don't hit HOME button or you will end up disconnecting.