Passive: Rebirth, when you die you will change into an egg and will be reborn in 6 seconds. This is a pretty useful passive, since it enables you to be aggressive early game and it prevents ganks on you if you have succesfully made it to your turret range.
Q = Flash Frost, this is your stun and initiation for a kill. This enables your chill effect on the target and thus hit more with Frostbite (Explained later in this post). Before level 6 you can only use this to enable your chilled effect but at level 6 you should ALWAYS make sure that the target is chilled by dropping R on them.
W = Crystallize, this will create a wall of ice where-ever you cast it. Use this to stop enemies from escaping or to prevent them from chasing you.
E = Frostbite, This is your huge burst spell, use it on your target every time he is chilled. DO NOT USE THIS ABILITY IF THE TARGET IS NOT CHILLED. That is the number one rule of playing Anivia properly. Only time you can use it when your target is not chilled is when he is at 100-200 hp and you cannot land Q or R on him.
R = Glacial Storm, This is an AoE damage slow that drains your mana over time. It chills your target and slows them. When your target is in this, that is the most easiest time to hit your target with E since it is reliable. Use this when you have succesfully landed a Q. Using this with in combination with Q and E will make a HUGE burst.
So in conclusion: Start with your Q, if it lands, quickly drop R on the target and then use E. This should take most of your targets hp. Repeat this everytime you can (It's shorter with my mastery tree choice) and you're golden!