Hi! My name is Kahare and I am a Lux main. I have mained Lux since Season 3 (2012), and have never lost a mirror lane versus another Lux! :) 
Here are my personal suggestions. I will put them in a list order so that it is easier to read.
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Your Skill Order should always be: E>Q>W (Maxing R when possible of course)! In fact, you should always take E first on Lux unless your team wants to do a level 1 invasion of the enemy jungle. The reason for this is because an E>AA combo is incredibly effective and can get the enemy laner down to 50% health before they hit level 2, all while keeping you at a very safe distance from them. There is no reason to snare them at level 1 since your follow up wouldn't be incredibly effective.
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Your damage combo will look like one of two things... the base is the same for both but the initiation is different.
a) [Target is exposed in front of minions] ----> Q>Place E (Dont proc it)>AA>E (proc it)>R(to finish them off) or another AA
b) [Target is behind minions] ----> E(Proc it)>Repeat until target is at about 40% HP> E(Dont proc it)>R to finish them off.
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#2 alluded to one of my biggest Lux tips ever. EVERYONE UNDERESTIMATES LUX'S E ABILITY! If you stay back and continuously E them until they are at 30% health with Lux's passive proc on them, your ult WILL KILL THEM. So if your enemy laner is playing it safe, it doesn't really matter. Just continuously E them and then when they run up the lane zap them with your ult. This takes some practice but if you master it you will do REALLY well with Lux in lane.
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Lux is meant to be played DEFENSIVELY. That means that you shouldn't die before the roaming phase. Naturally, Lux will have less kills than other members of her team but she will also have considerably less deaths if played correctly. If you DO NOT OVEREXTEND and play safely Lux has enough CC to escape from almost any situation.
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Build Order:
Starting Items - Dorans Ring, 2 HP Potions, Warding Totem (
,
x2,
)
First Back (Winning in Lane) - Build towards Morello, Boots of Speed (
,
)
First Back (Losing in Lane) - Build towards Athene's (This gives less offensive powers but gives more MR), Boots of Speed (
,
)
Core Items (Build almost always) - Sorcerers Shoes, Rabadon's, Zhonyas (
,
,
)
Situational Items - Luden's Echo, Void Staff, Liandry's. (
,
,
)
Notes: I almost always build Morello's>Sorc. Shoes>Luden's Echo>Rabadon's>Zhonyas>Situational Final Item
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Abuse Lux's ultimate skill! It actually recharges way faster than you would think it does and it is very effective. If you are leaving lane to back, use it to push your lane forward more! Also don't be afraid to use it to try and steal dragon/baron/blue when needed, but DON'T use it to steal kills from your teammates or they will not like you very much! :)
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Your W saves lives. Literally. If you see one of your teammates is struggling, use your Q to snare the chaser and try and get your W on them as soon as possible. Always throw your W into your entire team during a team fight. (This should be easy because as Lux you should be behind the rest of your teammates when the fight initiates)
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Pro Tip: Snare people under your tower, and the tower will kill them. It's brilliant if you ask me :)
Hope that was at least somewhat helpful! I love Lux with all of my heart and I think she is an amazing champion that everyone should play. Feel free to add me in-game (Kahare) and ask me any questions if you have them!!