LCK caster, Nick "LS" De Cesare, prescribes most players the Annie treatment, and it's for a good reason. She's a good champion which allows you to focus on everything but actually playing the champion.
Annie fits multiple roles. You can play her in Support, Middle, and even Top. She's going to be quite comfortable in all of them. Don't be discouraged by her ~48% win rate in Support: she's hardly a character that can lose a game from bot lane.
Instead of looking at the position, you should probably pay attention to matchups. Annie works best against melee champions, but she may struggle against the likes of Vel'Koz and Lux who match her range and out-damage her.
Her key gameplay mechanic is counting to four. Each used spell adds a stack. Get four stacks, and your fifth ability stuns your enemy (or enemies). This is one of the easiest core concepts to play around.
Once again, Annie's versatility comes in handy. A charged Q lands a guaranteed target stun, W may disable several people, and R is a game changer. And the spells still deal nice damage.
Annie's summon is one of the most intuitive ones. Sure, it's not a direct copy of a champion, like what you'd have with Shaco. However, Tibbers deals damage, gets faster when you get angrier, and can be controlled easier. Unlike Mordekaiser or Yorick, you don't have to deal with dozens of bugs.
Annie is less greedy than most AP Supports. She can still be effective even without stealing gold from teammates. AoE stuns will win you games even without incredible nukes off AP multipliers.