Ezreal is an amazing champion, and if played right, can steamroll over the other team.

Pros:
-Ezreal is an amazing carry. He has the power to turn the tides of the game.
-He has great burst damage, and his cooldowns are low enough that he can still spam.
-He is great at running away. His E ability is like Kassadin's Riftwak.
-He has a GLOBAL Ultimate, and can snipe very efficiently.

Cons:
-Very Squishy early/mid game
-Mana can be an issue, but not if you keep neutral buffs
-Early game he is vulnerable to ganks, as his E's cooldown is very long early on.

Early game is where you set the standard for late game Ezreal. Start in mid lane, or bot lane with a healer or tank, and last hit as much as possible. Whenever you can, harass with your Mystic Shot.

Wait for the right opportunity and use Exhaust/Ignite and pick up the kill in your lane.
This will give you an amazing early game advantage because your laning opponent will be scared from here on out.

Mid game, ALWAYS be looking in the other lanes for opportunities to snipe opponents with your Trueshot Barrage (R).

By this time you should have some kills, and be ready to start ganking.

Wait for the time where your laning opponent has gone back to buy, and gank some other lanes initiating with your E followed by Q.

Always make sure your farming and never go out of combat.

This is your baby!

Late game Ezreal is very powerful!
Team fights - single out a squishy champion and unload your cooldowns on him. Use your W ability to buff your teammates.

Always stay behind the tank - Ezreal is very squishy, and until his Frozen Mallet, he should be played very carefully.

Wards are what make Ezreal's ultimate matter. Place wards at Baron, Dragon, and down the river to maximize vision.

Ezreal's ultimate is an amazing stealer for neutral camps. Dragon and Baron are what you want to steal, and remember - It's all about timing.

Keep the neutral buffs on you as much as possible. Red and Blue make a huge difference in Ezreal's functionality.