Warning: Super long. TL;DR at end.
First of all, you definitely want biscuits, free boots, and cosmic insight in the inspiration tree. The biscuits give you better sustain and ~5 AD from Manamune. Manaflow and Transcendence in the Sorcery tree, Manaflow isn't strictly necessary in lane, but it gives you more AD again on Manamune, and when you get Muramana it gives %missing mana regen, so you'll get your mana back faster, and thus your Muramana damage goes up, since it's %current mana damage. Make sure your stat runes are both AD, no attack speed.
Second of all, Tear isn't a rush item anymore. Most lanes you want to buy at least a pickaxe or Sheen depending on your gold, then a tear. Some lanes, if you're snowballing, you want to actually just straight out get Pickaxe AND Sheen to keep up the pressure. Generally, you only do that if you know you can get a kill with Pickaxe Sheen combo that you wouldn't be able to get with just one or the other and a tear. Also, don't rush Lucidity boots and tear, then you literally just have 1450 gold worth of 0 combat items vs their BF sword. Of course you're going to lose.
As far as core builds go, start D-Blade, or D-Shield if they've got something like Sivir or Caitlyn where you're just going to get pushed in and farm all day, but if you've got a jungler you know who you can get to gank them for pushing, and have some decent hard CC, go blade to force kills. Then, just lane normally until your first back. If you're against someone you want extended fights against, you go pickaxe > tear > sheen > manamune. If you're against someone you need to play safe against and just poke/farm, go sheen > tear > manamune. If you're snowballing super hard and know your damages and know you can get a kill or two if you get sheen and pickaxe before tear, it's worth it, but it's very risky to do this.
Now that core's out of the way, we can move on to the rest of the build, which is entirely situational. If they have no tanks, you can go Iceborn Gauntlet into Luden's or Gunblade, then Gunblade or Luden's, then Deathcap/Spellbinder/Void. It got nerfed a little with the Q AP ratio going down, but it's still really good. If you're really ahead and none of their ADs can oneshot you, you can actually go Triforce Luden's Gunblade, but this is super expensive. The build also scales really well because AP Ezreal shits on squishies, and most people don't pick tanks. AP Ezreal also gets Sorc Shoes, not Lucidity Boots.
If they have one bruiser or tank, you can still go AP, but if your team fails to kill him and he doesn't kill himself getting kited by you, you're kinda out of luck, so especially bruisers with sustain, you'd want to go AD. 2+ Bruiser/Tanks and you definitely go AD. AD Ezreal is ENTIRELY situational after the core build of Manamune Sheen. If IBG will help, buy it. If you're against someone IBG isn't going to help against, like Warwick, (He ults and then follows your E with Q and a bigger MS boost than your slow, so he just runs you down.) then buy Triforce. Positioning is more important to deal with people like him and Singed. You want a lifesteal item after your sheen item. If they have an AP who's fed, or some hard CC, get Merc Scimitar. Even if their hard CC is easy to dodge, you can always just get hit by it on purpose and then cleanse as they commit thinking they've caught you. Big brain baits. If they don't have any of that, get BoRK 3rd.
Lucidity boots are your friend on AD Ezreal. If they have sustain, you'll wanna pick up an Executioner's Calling at your earliest convenience, but sit on it until you're full build unless they have someone with an armor item, components for another armor item, and tabi. Otherwise, you're kinda wasting your money since ~30-40% of your damage is going to be magic damage, anyway. Only build GA if you dying means the game is literally over. Most other situations you'll die in as Ezreal are not ones a GA will save you from.
If they have 2+ tanks/bruisers, you should honestly pick another ADC, but if you want to play Ezreal anyway, you either go Triforce if you can safely use the attack speed to get more bork procs, but if you can't you want IBG > Bork > Liandry's if you don't need any defensive items. If you have a hypercarry like Yi or Irelia or Jax on your team, they can honestly deal with those guys for you, though. If it looks like, for example, your Irelia is shitting on Darius, you can just go AP. You don't have to deal with Darius. But if your Irelia is losing, then you should probably build with that guy in mind.
Also, if they have 2+ bruisers and you don't need Merc Scimitar, you can go IBG > Bork > Liandry's, the slow + liandrys + damage ramp per second in combat makes you pretty good at killing tanks. This is all situational stuff, though, and you'll get a feel for when you should build what as you play Ezreal more.
Mechanically, make sure you're weaving an auto between every single spell if you can without fucking yourself. Start fights with your ult, it's not just a finisher. It'll chunk the entire team, especially when the enemy team starts going ARAM. Chunking all their health and giving you +50% attack speed, and their squishier members are likely half health or so, so they may have to just recall and give the objective, or overstay and die. Unless someone is caught with a CC or isn't in a position to dodge, don't W > Q. Typically, the W gives too much warning and they just outright dodge the Q because of it. If someone jumps on you, E auto Q auto W. If you REALLY need to keep escaping, just keep Q again until you can E. If you can keep kiting, auto between each Q and maybe chuck in another W for your next E.
Last, but not least, the gameplay. Starting with laning phase, hover bushes to fire Q's from so they have less time to react, and you're further from where the enemy jungler might come from. This also makes it hard for the enemy bot lane to play from the bush, since your Q is free as fuck. Stand just outside of the enemy's range. For example, Lucian dash is 425 units, and he has 500 Auto range. Stay 950-1000 units away, and you can still fish for Q's while he is unable to actually dash in and chunk you. If you stay 925 away, you'll have to E, and you'll still get chunked, and once you get chunked, it's kinda rough. Your goal is honestly just to survive until Manamune and Sheen, but that doesn't mean you have to play passively, just play smart. Staying just outside ranges, fishing for Q's, Q'ing minions to keep attack speed stacks up, then baiting the enemy support's skills is a great way to win the lane for yourself. If the enemy support whiffs, punish w/ a W > E auto Q auto trade on their ADC, unless their support is squishier. At that point, you can back off knowing they're chunked, or keep going and force them out/kill them, situational.
Keep an eye out on the map. Lots of times you can see people fighting, and if you see your teammate's health going down, then STOP going down without any enemies dying, you should check their lane to see if you can pick up a kill. Mid game, you really need to manage the waves so that the enemy has to respond to them before they have a chance to just group on your team because you're too busy shoving a wave. Ezreal's waveclear sucks, so you have to be mindful about doing this ahead of time before the enemy can capitalize, then just leave and join your team and contest towers/objectives with them. If you know for a FACT you can get a tower, ult the enemy minion wave as close to the tower as possible and use your attack speed, then reset your attack speed bonus w/ W on the tower. If your team dies in the process, it's on them. Anyway, mid game IS when you're at your strongest. Use this power and group with your team unless you guys really can't win a 5v5. If you can't win a 5v5 because of your own shortcomings as an Ezreal player, honestly, just force it anyway. You'll learn sooner or later. If you can't win the 5v5 because one of your teammates is just not doing their job, you're kinda forced to split until you get vision on the enemies trying to rotate to you. If you can engage them without dying, do it. I don't mean if you can kill them, I mean if you can even Q them once and not die for it, just do it. They'll get greedy and try to kill you, and you'll be able to waste their time. Or they leave and you can keep chasing them to see if you can waste more of their time, to keep your team from being outnumbered. Obviously don't do this if they have a shot at running you down and killing you, but if it's safe, just waste their time then back off. If you see them regroup mid, keep splitting and rinse and repeat. Most people can't 1v1 you mid game unless you fuck up, so they'll eventually just send two for you and your team can probably win the 4v3.
Lategame. If you got to late game as Ezreal, though, chances are you did something wrong. You DO get outscaled by most ADCs, except you can still pretty much oneshot them if you catch them out and land everything, so it's not that bad. . Even as AD. AP can just straight up W E auto gunblade most squishies and AT LEAST force a back, even without Q or ult landing. I've killed squishies with just W E gunblade before. It can get pretty nuts. AP kinda plays like an assassin, because you have the potential to make picks. You can also, again, ult down mid when the enemy team ARAMs it and they can't contest mid lane anymore if you hit at least 2 or 3 of them. AD has to play a bit more safe and less aggressive, but it also has far more defensive options. If you're all your team's kills and they have someone who can actually get on top of you, just go AD and build against that person and make sure you don't die, unless you can just go AP and oneshot the person. Late game, a lot of the impact you're going to have is mostly just from not dying and chunking people with your ult to force them to defend something with low HP or just recall and give it up
TL;DR Buy Doran's Blade at the start, don't buy tear until after Pickaxe or Sheen, throw autos between spells, don't W Q warns enemies about Q, Manamune IBG Bork is the best beginner build. Lord Doms is oft bad. Land Q to reduce CD on E
