As Ezreal, you are going to have to come to terms that you are at a disadvantage in an even skill matchup against most ranged carries. You cannot reliably hit your Mystic shot in a minion wave, your Arcane shift will hit your unintended target and your Essence Flux is quite low-damage. He's probably the most snowbally AD-Carry in the game due to high base damage abilities, mixed damage, long range, a blink and a large attack speed steroid.
-Look up various Ezreal builds and find one that can fit your playstyle. My personal favorite is Manamune - Trinity Force - Boots of Lucidity - Blade of the Ruined King - Last Whisper - Infinity Edge. Pure damage.
-Find good opportunities and combos to trade with your opponent. If you have Sheen, use Essence Flux on an enemy champion if you cannot reliable hit your Mystic shot and auto-attack them for a Sheen-proced auto-attack. You're more likely to win a trade this way.
-The general rule for poking is to poke whoever you can. As you play AD-Carry more, you'll pick up when someone is out of position, which direction someone is likely to juke and how much damage you can output vs. them.
-Freeze the lane if you have a surefire advantage where you can fight effectively through a minion wave and zone them from farm. Push when you are trying to gain an advantage. Pushing ensures that you have more minions fighting on your side for you when a trade occurs.
-Coordinate with your Support. Ezreal can work well with any Support as long as you coordinate with pings, words or you're both just on the same wavelength.
Good luck Explorer.