Ezreal is one of my favorite ad carries in the game. obviously you should be bot with a support champ, alistar being probably the best just because they cant doge your q and w. he does great damage early and if fed, can dominate mid game and end the game before he dies off later.

Pros:
Very good damage early-mid game
Fun to play
Can pull off cool tricks with his E

Cons:
Very squishy
Hard to play if not good with skill shots
Damage dies off later

I chose Flash because it has saved me so many times. If you slip up and use your Arcane Shift to get into a battle or it doesn't bring you far enough away, you can be in trouble. The cooldown can be somewhat long for some of the later team fights and if you're getting away from a gank.
I also chose Exhaust because its a great spell to have for an ad carry. it helps in 1v1's and against their ad carry

These items help to maximize your damage with q and autos with sheen into trinity. DON'T rush trinity force. Its a great item but its not that great to rush because its not a damage item. you do need the sheen, however, and all the arpen from runes and items makes you hit ridiculous hard.

In team fights you HAVE to stay in the background. Your auto attack is the most powerful and reliable tool you have. Your Q is your hardest hitting besides your ult but it wont always hit your target. Definitely be firing all your abilities into the fight and your ult is best used right after the initial burst of damage to be most effective. Try not to use your E until the fight is over and youre chasing or running away.