This is how I play Ezreal it has a few simple suggestions but nothing that is going to give you the ultimate secret to playing Ezreal amazingly. Learning how to play Ezreal takes time just because he is all skill shots.

AD Ezreal

9 Marks of Desolation. I use these because the armor penetration is important on any AD character.

9 Seals of Clarity. If you cast Ezreal's Q even half as much as I do you will still run out of mana fairly quickly and these help to combat that early game.

9 Glyphs of Celerity. The cooldown reduction makes Ezreal deal higher damage if you like to poke enemy champs with his Q a lot. Having these glyphs just makes it so that you have more chances to hit them.

3 Quints of Desolation to add onto the already high armor penetration that the marks give.

AD Ezreal

Having 21/0/9 gives Ezreal some extra damage toward the enemy champs and towers. Also if you ever get a buff from teh enemy jungler or just feel the need to get one yourself it's nice to have it for just a little bit longer.

I'm still figuring out what I like to use as items for Ezreal so I will update this in the future.

AD Ezreal

With AD Ezreal it is important to level up his Mystic Shot first because that is one of the main damage dealers with an AD Ezreal. Arcane Shift is more of an escape or chase move when you play with AD Ezreal. Essence Flux is almost useless with AD Ezreal but it is nice to have if you are laning with someone to use it on them and keep your passive stacks when there is a minion wave incoming but not close enough to hit.

Hybrid Ezreal

Hybrid Ezreal you start off with your Q because it helps you farm and is the cheapest of his skills that can actually hit minions. It is a great tool for farming and harassing the enemy that you are laning against. Grabbing an E next is also nice because it is a great escape or chase move that can deal a bit of damage to your enemy even though you wont have much AP in the beginning. I only get 1 in Ezreals W until around level 12 Because that is usually when I get around to buying my Archangel's which is where the majority of your AP is coming from and by then you have a fair amount of mana letting you spam Essence Flux through the enemy minions to just poke your enemy.

Ignite is one of my favorite summoner spells because it is great to take down a low health enemy champ if they are under their turret.

Flash is just amazing with Ezreal. It gives him the ability to Flash twice basicly meaning he can get away from almost any situation.

Ghost isn't a bad choice if you end up falling behind the enemy extremely often.

Clarity isn't really a good choice for Ezreal. It doesn't help him in combat but it can be nice if you need to see where the enemy jungler is or to check if Baron is being taken.

Exhaust is nice to have with AD Ezreal slowing the enemy letting you land a few more auto attacks on them before they get away makes it almost impossible for them to escape.

Cleanse is okay but it just doesn't seem worth it to have with Ezreal because if you need to escape you already have Arcane Shift which you can use to get over a wall or past an enemy champ.

If you play extremely aggressive Ezreal then heal might be a good choice because you can tower dive at two hits and get the enemy champ to come at you rather than run away because they think you will die to the tower.

Teleport is nice if you recall a lot and don't end up staying in the fights long because you run out of Mana or are focused.