You don't have to complete a Pokemon contest on any version; just step inside the building for the first time to see the dialogue scene and advance the storyline. If you're playing Platinum, your next step is to challenge the Hearthome City's (that's the town you're in) Gym Leader, Fantina, who uses Ghost-type Pokemon. If you're playing Pearl or Diamond, take the exit out the South-East corner of town to Route 209, which will take you to Solaceon Town.
A shiny Pokemon is a Pokemon whose color is different that it would normally be (ex. a red Gyarados is a "shiny" Gyarados, as Gyarados are usually blue). They're very rare, but other than their color, there's nothing particularly special about them (at least as far as I know). I think there's a shiny sparkle, of sorts, that pings next to the opposing Pokemon when you enter battle against a shiny Pokemon (not entirely sure about that). When you own a shiny Pokemon you'll notice there's a red star next to its portrait on its character summary page. Chances are, though, that you won't find a shiny until after you've beaten the game.
