Flash & Heal are going to be your go to summoner spells, There is no need to replace Flash what so ever and Heal is amazing. Both these spells go hand and hand in finishing off a kill and gank escape potential
This rune page will get you to 40% CDR by the end of the game
The page above is going to be the average page for many adc's and I find it perfectly fine on ezreal.
The blue build does benefit from "Oppressor" allowing to abuse of the bonus 2.5 extra dmg, for the more experienced/confident carries Bounty Hunter does allow a higher overall dmg out put. (Once you get at least 3 unique champion kills 3%>2.5
R > Q > E > W
Rising Spell Force (Passive): Hitting a target with Ezreal's abilities increases his attackspeed by 10% for 6 seconds. This passive stacks up to 5 times which gives a whopping 50% attackspeed increase, it's probably one of the best AD passives in the game. Make sure to stack it up by using your ult to hit multiple enemies/creeps or activating W to your allies.
Mystic Shot: Ezreal fires a skillshot that deals physical damage and applies on-hit effects, the cooldowns are reduced by a second if it hits a target. This is what makes or breaks a pro Ezreal player. This can be used to zone enemies in lane or used after an auto attack to boost your damage over time. It also comes in handy in lowering your cooldowns especially for your E.
Arcane Shift: Ezreal teleports to target location and deals magic damage to the nearest enemy. This is like a free flash with extra damage. Max this 2nd as it gives more damage and lowers CD to almost have it always up (Q cd reduction). Ezreal's E can be used to cancel abilities like 53_icon_64.png hook 27_icon_64.png fling.
Essence Flux: Ezreal fires an AOE skillshot that does magic damage and increases allied attack speed. This has been nerfed recently and doesn't slow enemy attack speed anymore. This is why lvling W is somewhat useless and makes you mana hungry. Use this to poke or add up your passive stacks.
Trueshot Barrage: Ezreal channels for a second and fires an AOE skillshot that deals magic damage and deals less after each unit hit. This can be used to snipe people across the map or used at the beginning of a skirmish to add up your stacks. This also reveals enemies that are hit and comes in very handy during Baron.
> > > > >
Doran's blade and a pot no questions asked.
There are two pathways for Ezreal, Trifroce > BOTRK or Manamune > Triforce. I personally prefer TF > Blade but play around with it and see for yourself! Manamune route is a lot safer in lane because of free Q's to cs but takes a while to ramp up, TF 1st on the other hand has an earlier power spike.
Frozen Fist works well against bruisers as it gives an AOE slow, CD reduction, and armor to kite them for days.
Asian Ezreal build has been proven legitimate by the asian scene, so try it out and have some fun!