Playing only AP Ezreal gives you a lot of burst and poke with your W, it also gives a large amount of damage with your Trueshot Barrage, AP Ezreal is commonly played mid.
Playing only AD Ezreal is good for poking with your Q but is mostly used for auto attacking and pushing, commonly 'blue build' Ezreal is used, AD Ezreal is mostly played bottom.
Hybrid Ezreal gives a nice balance between the 2, giving large amounts of balanced poke between basic attacks, Mystic Shot and Essence Flux, hybrid Ezreal gives the largest sustained poke and also gives a range of damage types, both Physical and Magical making you hard to itemize against. Hybrid Ezreal, built AD will allow you to push like an AD Ezreal would but give your W increased damage. Building more AP hybrid will allow you to have slightly less burst in exchange for giving more DPS through basic attacks.
Hybrid Ezreal gives a larger variety of possibilities throughout the game and increases the amount of possible plays.

Flash is a must have, the amount of times it could save your life is insane, honestly, Arcane Shift and Flash give a huge distance between you and the badies.
The summoner spells I could recommend from experience are;
Ignite- For increased kill pressure in and out of lane.
Barrier- Against burst champions, such as Le Blanc.
Cleanse- When playing against certain champions, such as Vi, Warwick and Sejuani.
Ghost- Not as useful but if you plan on roaming a lot, ghost can be useful.

In lane, if you have chosen to get an early tear you should be spamming both Q and E on the way to lane, as this means you can stack with the lowest cost spell and get to lane quicker using Arcane Shift. With a tear you should also be harassing the enemy laner with your W in order to try get them to a low health. By getting your opponent to a lower health you will be reducing the kill pressure they have and increase chances of your jungler ganking, which of course, is good.
If, in lane, you have chosen to go for an Aether Wisp in order to get an early Luden's Echo, you should also be trying to poke the opponent down with your W but not try and auto trade as well, as waiting for the animation may give time for your opponent to trade back.
Choosing a sheen in lane will allow you to easily poke with your Q assuming your opposing laner is not aware of the damage it will do. If you get Sheen in lane you should be trying to get a Trinity Force as soon as possible to reach an early power spike.

Basically, in lane you should be poking down the opposing laner in order to achieve a more dominant position and you should also be trying to farm well, obviously.

When a fight breaks out near the mid lane you should be able to get there quickly, although keep an eye on how much mana you have left as early on having no mana can leave you completely useless in a fight. If a fight occurs in the middle of the jungle E.G. Near dragon or baron, you should use Arcane Shift to close a distance whilst running, if you do this as soon as you see the fight you should be able to get there in time and also have Arcane Shift off cooldown before the fight ends, giving you a lot of mobility.

The items you have at the time can largely impact your role in a fight. For example, if you only have a Luden's Echo you should be trying to line up a good ult and use your W whenever possible in order to do damage. Also using your Q in fights is very important after using your W, this is because if the Mystic Shot hits a target it will reduce all of your other cooldowns by 1 second which can mean a lot when your W is mid cooldown.
Another item example would be Trinity Force, if you have Trinity Force and no other large items you should be mostly poking with Mystic Shot and auto attacking, lining up a good ult will also be useful but auto attacking as much as possible could make a large impact.

Basically, in fights you should try to position yourself to hit a large impact Trueshot Barrage whilst constantly using your most damaging abilitie(s).

To end the game as Hybrid Ezreal, you should either aim to fight or siege, depending on your items. If you have gone for the more AD based build, you should be aiming to make picks or force a 5v5 fight. This is because your basic attacks can do a lot of damage to tower, inhibitors and the Nexus. As AP Ezreal you should aim to poke down the opposing team in order to get a successful fight, this will allow you to push, alongside the ADC in order to win the game.

This is my first ever guide, any feedback is welcome as long as it's something I can improve on, such as spelling misteaks or words order confusion.