Welcome, we all know Ezreal as that awesome AD Carry that everyone loves.
I've never played Ezreal as an AD Carry.
Ezreal is a prime AP Carry that can be suited to mid-lane due to his long range pokes and mobility, not to mention his persistent cooldowns that keep him putting out damage. To me, he is best AP due to all four of his abilities scaling with it and the ridiculous long range on every skill. Although his damage and utility are definitely not at the top of the League, he can easily snowball hard by being an aggressive assist with DPS and ganks. His real strength comes with the ease that he can poke enemies, wear them down, and unexpectedly surprise them with his burst.
AP Ezreal is a bulldozer--slow, but surely powerful.
AP Ezreal is a
Mystic Shot is a great skill that is perfect for two things:
2) Poking enemy champions
3) LOWERING YOUR COOLDOWNS
Despite its low AP ratio, it really shines early game with #1. It can help you last-hit with just one point, and by the time it can't be used easily anymore (around level 6 or so), you should already have Sheen. Later in the game, Lich Bane will turn this into something that really HURTS, but Lich Bane is not necessarily an essential item until late game.
MOST IMPORTANTLY--SPAM THIS ABILITY IN ORDER TO KEEP LANE PRESSURE AND REDUCE COOLDOWNS. (Mana problems should be solved by an early Eliesa's Miracle and mana regen/18 runes.
Essence Flux is THE main poking device for early-game. You will max it first. The raw damage that kicks in around level 3 is what really will deter enemies, leaving you to farm with Q and E. By the time that the enemy can really push hard, you will have your ultimate, which will easily clear the lane. Stacking AP earlier than this guide suggests will simply hurt your enemies more.
Keep this on low cooldown by using your Q.
Arcane Shift is basically a Flash + homing missile. The missile is ignorable in many respects, but can help you to last hit. When you are relatively safe, you can use this ability to push back from your turret. NEVER USE THIS WHEN BEYOND THE MIDDLE OF THE LANE. During the early-game, you WILL be ganked if you pushed up to your tower. Like real life--SAFETY FIRST.
That being said, it is a very helpful tool for ganking bottom/top lane, and for killing recalling enemies by the tower.
Arcane Shift is not Flash--thus, you have more freedom in using it. However, make sure that you can use this AND Flash in a sticky situation. It saves your lives.
THIS is the skill that truly makes AP Ezreal powerful. Use it constantly, but save it for during/after team fights. It solves Ezreal's pushing problems, and shooting it down the middle of the lane results in natural lane pushing (you will hit more than one minion wave, ensuring that your full-health minions will kill them faster). When dueling, use it when going in for the kill in tandem with your E, and if you are in danger of dying in a team fight, use it before you die. It will take off a large chunk of health off anyone (minus tanks), so it is essential that you use it right.
Don't be afraid to use the ultimate to clear minion waves and "ganking" other lanes. With your Q, it can be effectively on a 30-second cooldown.