Hey I'm diamond player on EUNE.
I have been playing as an ADC main since S3.
I ended S3 as a Silver, S4 as Gold, S5 Plat and now I'm in Diamond :D
Diamond 3 and promotion matches to D2 was the highest elo I've ever been.
I'm not the strongest mechanical player but the one with quite a big amount of knowledge.
I will try to explain you as much as I can and not hurt english language that much.
Have a nice reading :D

Pros

+ Decent in every matchup + AoE Global ultimate + Poke + Self-peel + Huge dmg in mid game + Really safe pick + Blue Ez is even safer + AS sterid from passive

So Ezreal is kind of adc that can do well in almost every matchup. Also his ability to self-peel ( Arcane Shift) is great for soloQ where you are almost always left alone with no help. Your ultimate gives a lot of possibility, you can set lanes on the other side of the map to push with it or just grant you fast stack for your passive before teamfight.

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Cons

- Low pressure during Laning Phase - Spells are skillshots - Falls of in late game - If behind most ADC outdamage him

Okay, so your laning phase is safe, that's without any doubt but against good players who can manipulate wave and just use brain and stand behind minions, your only one dmg source is autoattacking. Q, W, R and even E are SKILLSHOTS, you have to hit with them to deal dmg, especially when your main DMG comes from Q. Another thing is that you fall a little bit in terms of doing dmg in later stages of the game but you still do a lot to squishy targets so it's not like you are useless in late game but you can't solo carry then (in mid game you can). And the last thing - being behind is one of your main targets on lane, you HAVE TO be EVEN or AHEAD to be usefull when first turrets go down and you start grouping, otherwise your team will be playing without adc and you will miss the part of the game when you are the most powerful.

Let's be quick

1. don't level up any spell until 1:40 you never know what will happen and when enemy decide to invade levelling your Arcane Shift will be much more worth than burning Flash.
2. generally you want to start with Q as it gives you dmg, has low mana cost and low cd
3. so: R > Q > E > W, only one thing you can change is maxing W on lvl 3 and not giving second lvl to your Q, but it's up to you :)