Ezreal can become very mana reliant if you use your w or e too often. Try to only use your q in lane. Using Essence flux and Arcane Shift too often will drain your mana after just a few uses, even after getting a tear. When laning, most other champs can beat you in a straight on fight. As Ezreal it is very important to poke the enemy champs as often as you can, to prevent them from winning their engages. Ezreal works best with either strong poking supports, or good healing supports.
Personally I prefer having either Soraka or Sona as my supports. Sona and Soraka both have good healing capabilities, and can help keep your sides hp higher than the opponents, making a gank from your team much easier and safer. However "nontraditional" supports can sometimes work greatly too, so long as they can poke or have a strong engage/disengage.

Essence flux as stated previously is Ezreal's greatest ability. But when exactly should you use it?

When your ult or arcane shift is down, you should use it whenever you can hit something to lower those cooldowns, and get them back up.

While laning there are really just two things essence flux can be used for: CS or poking an enemy champ. If you are up on CS and want to engage the enemey or prepare for a gank, missing some CS to poke the enemy may be the right call. However, if they have good heals and/or shields up going for CS is always way more important than doing little to no damage.
When in lane and not setting up for a gank or engage, i would personally say that using q to secure CS is more important than hitting the enemy ADC or Support.

When split pushing, using your ult to clear the wave and get out of an unsafe position, or quickly get back to your team is usually the right call.
If you are good/lucky with timing, Trueshot can sometimes grab a kill on the other side of the map, or even snag a buff or dragon.

When in lane: Its important to stay safely and away from the supports if they can CC you, especially when arcane shift is down. Its also important to angle yourself in a position where you can hit behind the enemy minions to poke them down if needed, but still be able to q or auto hit minions for cs.
When in team fights: Ez needs to be near the back line in order to stay safe, since he is very squishy. not to say you want to be out of range to q, because you might be the one to catch someone with your iceborns slow. Its also important to stay near your Attack damage based champs to hit them your w for the buff.