Hi,
I want to present you my way to play this awesome champion.
I also added a build for those who are not that trained with him and who just want to try him out.

For normal i take:
Marks: Armor penetration
Seals: Mana reg. per lvl/ health per lvl for untrained
Glyphs: Magic resist ( very usefull against any casters in mid)

I take 9/0/21 in both builds cause you need the cdr and manareg. to spam your skills for farming and harassment. Cause even though this is a guide for AD Ezreal, you are not a true auto- hit champion, you cast your spells most of the time.

For the trained ones I suggest to start with a Regrowth Pendant.
Port back as soon as you can buy Philosopher's Stone and Boots of Speed. It will help you getting money very fast and it keeps your mana up.
Then Rush the standards like Trinity Force( this is by far the best damage item for AD Ezreal, cause of its perfect synergize with Mystic Shot throgh the 150% bonus damage), Banshee's Veil( you need it cause if you play correctly, you should be one of the highest damage dealers in your team and always focused), Bloodthirster+ Blach Cleaver and Infinty Edge( those three items will get your damage explode).

Now for boots. You truely can buy all boots EXEPT Berserker's Greaves and Ninja Tabis. You neither need the attack speed as Ezreal cause you have your passive and with Trinity Force and Black Cleaver you have enough. Tabis are mainly for melees and tanks, you really don't need them.

I use Teleport and Ghost.
Teleport cause I can port back to get my Philsopher's ASAP and to push. Remember you are playing a ranged AD carry. Later on you can shred towers in seconds. Make use of it. If you see your team ' playing' with the enemy team on a lane, just port to the lane where your creeps are nearly at a tower and push it. Enemies will hate you!
I use Ghost and not Flash cause you already have a flash: Arcane Shift! You are Ezreal, you are Flash! :D If you dont waste it for senceless initiates and chases, you can nearly always escape a gank.
Sure you can use other good spells like Exhaust, Ignite, Flash( yes you can use it if you are such a fan of double Flash, but i think it's a little waste).
Never take Heal or Clarity...WASTE

As AD carry you need mid lane or if you have to go duo, the go with a support to ensure nearly all last-hits to you. You need the money ASAP cause Ezreal is a strong early/ mid game champion but to get him strong in the end he needs tons of farm or many kills( but you really cant rely on killing enemies, cause if they are just a bit smart they know you strong eraly and play defensive).

This is not a must guide, it's just my way of playing Ezreal, who is with Akali my favorite one :D. If you have ideas and suggestions, just tell me. AND PLS TRY IT OUT BEFORE VOTING DOWN!!! THX
And sorry if my english is not correct everywhere! :P