Hi, Euc here :). Well, this is my first ever League of Legends champion guide, so don't be too harsh, but constructive criticism is always appreciated ^_^.
I've always loved playing Lux since I bought her. She probably isn't good for beginners because all 4/4 of her skills are skill shots (you need to aim them); she also doesn't have much HP compared to other champions throughout the game, so you really need to know when to back down. Enjoy...

PROS
EXTREMELY long ranged
High damage output
Great sniper
Low cool downs with Runes/Masteries/Items

CONS
Squishy throughout the whole game without health times
Focused/Ganked if you do well
High cool downs without Runes/Masteries/Items (which is why I don't recommend Lux to new players)

I used Greater Mark of Magic Penetration marks because of their magic penetration, which is pretty self explanatory.
When it comes to seals, you can either use Greater Seal of Scaling Mana Regeneration or Greater Seal of Replenishment. This depends on your playing preference. Once you reach level 7 (which if you're mid, should happen fairly quickly), Greater Seal of Scaling Mana Regeneration will be better than Greater Seal of Replenishment. So pretty much, Greater Seal of Replenishment is more worth it to me.
I use (or will use once I can afford them) Greater Glyph of Cooldown Reduction to decrease the cool down on most of my abilities. Lux is pretty much useless without her abilities, so shortening the cool down of her abilities is crucial.
Greater Quintessence of Health give Lux more HP, which will really help you survive early game.

I go with standard 9/0/21 caster masteries for shortened cool downs, magic penetration, and mana regen. No point having anything in defense because you should always be staying behind your team. Anything more in offence is more AD (attack damage) oriented, but you're going all AP (ability power). If you decide to get Ignite, it would be good to get the point in offence and go 10/0/20 (removing a point in Flash or Ghost depending on which one you decided to keep).

WHAT YOU SHOULD GET
Ghost-Great for running from bad situations or getting the speed boost you need to get an enemy in range of one of your attacks
Flash- Also a great escape tool. Always keep in mind Lux's squishiness
Cleanse- I don't think I've ever seen a Lux take this, and I think there are better options, but it can still help her get away from heavy CC
Clarity- If you're a lower level without the Runes and Masteries for mana regen, this is a good spell to have since Lux is useless without mana
Ignite- Also a good option for more damage, to counter Master Yi's Meditate or Dr. Mundo's Maximum Dosage
Clairvoyance- is also an option. You are part support and it also helps spot low health enemies so you can finish them off with your ult
Teleport- Good for traveling around the map and in case you're just lazy :P
Fortify- Again, you are part support, and this can really help save your towers. Snaring while an enemy is in range and using this can almost guarantee a kill
Heal- Gives that extra health you need in a bad situation and heals your allies as well

WHAT YOU COULD GET
Exhaust- This decreases damage output and slows, which could be good for getting a kill or running. But remember you also have your own snare and slow.
Rally- Increases AD, and since you're AP this is useless for you. But if you get the mastery point for it, it will increase your AP as well. If placed strategically and if it's not kill, this can provide an advantage for your whole team.

WHAT YOU SHOULD NOT GET
Revive- You should only get this in VERY SPECIFIC SITUATIONS, Lux pretty much never needs it.
Smite- Even at level 18, you're still a terrible jungler, absolutely no reason to get this

I always start out with a Doran's Ring and head to mid. Laning top or bot with a stun is also great. An example is Sion. If Sion can stun, you'll have no trouble landing your Light Binding Light Binding. On my first trip back I get Tear of the Godess, which is extremely good on Lux, I get it every game; it gives more mana, and more mana regen. After that I get [Ionian Boots of Lucidity] for the cool down reduction. Sorcerer's Shoes is also an option, but they are more expensive and I like to spam my abilities, including my ult.
In most games, I get Rod of Ages next, but if I'm having a really good game (I'm getting alot of kills and almost no deaths), I get Mejai's Soulstealer instead, for the increased AP and cooldown (at 20 stacks).
Next I get Rabadon's Deathcap for the nice AP increase. Next item is Deathfire Grasp for the cool down reduction and AP bonus. I like saving its active for tanks >:] since it'll do the most damage on them. If the enemy team has a lot of DPS (damage per second) champions like Master Yi or Jax, you should get Zhonya's Hourglass for the armor, two second invulnerability active, and AP.
No Tear of the Godess is complete without upgrading it to an Archangel's Staff, which I usually do last, but you can do it earlier if you want.
You should always sprinkle in an Elixir of Brilliance here and there.
Remember that buying items is always all situational and you should look at your enemy and how good you're doing to decide what items to get.

I get Light Binding first for its snare, the damage isn't very important to me. I max out Light Binding last. This spell is great for setting up for your Lucent Singularity. It is also great if you're being chased, just throw it behind you, pop Ghost, and run.
At level two, I get Lucent Singularity. This is your main harrasing spell and should be used as often as possible. One thing I love about this skill is the vision bubble that surrounds it. It's great for scouting bushes, and it lasts 6 seconds unless you detonate it. You can shoot this spell behind walls, which is great for peeking behind walls and ulting people to finish them off.
At level four I get Prismatic Barrier and max it out shortly after Lucent Singularity. This spell is also good for escaping a bad situation, simply shoot it in front of you and keep running. It gives you and all teammates who touch it, up to two shields to absorb damage.
Ahh, Finales Funkeln, which means "final sparkle" in German in case you didn't know. This is commonly referred to as her laser. (Press /j three times and she'll say "shh, I'm charging my laser (: ). This might be my favorite ult in the game. Unfortunately, it isn't global, but it still has some good range (probably from where you spawn to the inhibitor turret, give or take). I always watch for low health players who run on off and just finish them off with my laser. That's probably the best time to use it, you shouldn't be using it at the start of a team fight. Illuminating the target with either Light Binding or Lucent Singularity will cause Finales Funkeln to do more damage, depending on your level.

Laning Mid- I usually like to take mid for myself. If you've ever been mid, you know a good Lux is extremely annoying and even deadly. Once you get Lucent Singularity, you should be spamming it whenever you can. It's a great harassing skill because of its range and damage. Just throw it at the enemy and press E again so they don't have time to run. You can also throw it on one of the two bushes on the sides to check for a gank. Only use Light Binding if you have a clear shot, and right after it use Lucent Singularity and pop it. If you need to go back for whatever reason and your enemy is still in lane, a Finales Funkeln should make them go back to their base.

Laning Top or Bot- If you don't have Clarity, it would be great for your laning partner to have it. Another good thing for your laning partner to have is a stun. Sion with clarity is an amazing laning combination. If you're a good Lux and your partner is a good Sion, you guys should dominate. As for how to play, it's pretty much like in mid. Focus your harassing on one of the players and once you get him or her low enough, go in for the kill. If you can't get a good partner, you should go mid, or do with what you have.

Laning Solo Vs. Two Most good teams have a jungler, and if that's the case, Lux is pretty good vs two. You should never overextend, you should stay by your turret and defend it. Just farm and harass until your jungler is ready for a gank. Just snare and watch your team kill score go up.

My most important advise in team fights is to STAY BEHIND YOUR TEAM!!! You are very squishy and if you are by yourself you are pretty much dead. Lux]'s abilities are very long ranged, so you're a lot more useful if you are behind your team. Keep an eye out for enemies on low health. I always check their health so I know that using my ult will get a kill.
You should also keep an eye out for enemies who are MIA (Missing in Action) because if you get ganked and you aren't ready for it, you'll be an easy kill. Remember to use Prismatic Barrier to help keep your allies from being killed.

I'll add more as I go along. Note: In most of these games I didn't completely follow my own guide. Like I said before, it's all situational ;)

That's about it. Remember to comment and rate me. I am happy to receive advice from you guys. Credits go to vkluer for giving me some extra tips, and other Lux guides who gave me ideas and helped me get started with Lux. My in game name is Euc, so add me if you want. Thanks for reading!