This is a guide to AP Lux mid.

Pros
Great Range
Excellent at Zoning Enemies
Long Range Ult
Really Good Snare (hits two people!)
Lots of AoE damage

Cons
Squishy
No escape if you miss Light Binding
Terrible base mana regen
You MUST be good with skillshots

What does Lux do?

Lux is primarily a mage-support hybrid. She can easily set up kills for the rest of your team with her Light Binding However, with enough AP, she can easily melt the enemy team with one laser.
All of her abilities have decently long cooldowns but extremely long range. As such, Lux is all about harassing and attacking from afar or getting your entire burst combo down and running away. You do not want to get too close to them where they can counterattack.

The standard Lux combo is Light Binding into Lucent Singularity into finales funkeln into Ignite and an autoattack.

Teamfights

Stay behind your team and abuse your massive range. Try to poke the enemy team down with Lucent Singularity. Slowly whittle your enemies HP down. Try to land a Light Binding on them if they're out of position. It can set up your allies for an easy kill and easy fight.

Once you get into a teamfight, throw out Lucent Singularity where they're most concentrated. If anyone tries to go for you or your carry, throw a Light Binding to stop them. Immediately use Finales Funkeln and try to hit as many enemies as you can. Use Prismatic Barrier and Ignite Start autoattacking while you wait for your cooldowns.
Rinse and repeat. If you have a lot of CDR, you may be able to use your ultimate 2 times in one fight.

Recommended

Flash Flash is an excellent summoner spell used for escaping and repositioning.

Ignite Ignite is useful for that last bit of extra damage to kill them.

Teleport Useful to get back to your lane quickly or teleporting onto a ward at bot and getting a double-kill.

Clairvoyance CV isn't used much anymore but it can be useful to see where the enemy is for easier sniping. You can also use CV to help your jungler counterjungle them.

Blue Buff and increased mana regen.

Magic Pen reds Greater Mark of Magic Penetration To hit harder.

Mana per level Greater Seal of Scaling Mana Regeneration Lux has terrible base mana regen. More mana helps.

AP per level Greater Glyph of Scaling Ability Power To hit harder.

Flat AP Quints Greater Quintessence of Ability Power To hit harder.

Standard build is

Boots + 3 Health Potion

Double Doran's Ring

Sorcerer's Shoes

Rabadon's Deathcap

Situational

Situational

Situational

If you're having trouble farming, you can buy a Kage's lucky pick for the GP10 and turn it into a DFG or Morello's later.

For situational items, you can use

Void Staff

Athene's Unholy Grail

Banshee's Veil

Abyssal Mask

Guardian Angel

Deathfire Grasp

Lich Bane

Morello's Evil Tome

Or whatever else catches your eye. They're situational items after all.
Don't get items that would negate each others' usefulness. For example, don't get Morello's Evil Tome if you already have a Athene's Unholy Grail. Don't get Abyssal Mask if you get a Void Staff.

Remember to buy a Elixir of Brilliance after you get a major item or before any major teamfight. It might not seem like much but that extra AP and CDR really do help. With just blue buff, an Elixir of Brilliance and the 4% CDR masteries, you have around 34% CDR. That should be around a 26.4 second cooldown on your laser.

Archangel's Staff

You have long cooldowns. You won't be able to charge up a tear very fast and it won't be worth your money. Get a chalice and turn it into Athene's Unholy Grail if you have that many mana problems.

Rod of Ages

RoA, in order to be useful, needs to be your first item so the 10 minute timer can pass and it will achieve maximum potential. If you get a Rabadon's Deathcap first, then it'll be too late to get a RoA. The game will be pretty much over and RoA would not be worth your money. If you get it first, then you won't have enough burst to kill anyone.


Haunting Guise

Sorc shoes should provide enough MPen for early game. If you need MPen that badly, get a Void Staff. Even if they're Galio or stacking MR, please don't get this. It doesn't build into anything and doesn't have much worth.

Mejai's Soulstealer

Unless you're absolutely stomping, please don't ever get snowball items. They're bad in general.

Ionian Boots of Lucidity

Ionian boots of Lucidity just aren't worth it unless you are blue-deprived for the whole game. Sorc shoes would give you a much stronger early and mid game and if you want CDR that badly, you can buy a Blue Elixir.

Play passively. Last hit the minions and don't spam your Lucent Singularity. Once you hit level 3 feel free to begin harassing your opponent. When you hit level 5 you can kill the caster creeps with one Lucent Singularity and one auto attack each. Continue harassing your enemy with Lucent Singularity Don't bother using Light Binding until you get your ultimate or your jungler comes.

When you hit 6, try to land a Light Binding When you do, then quickly follow up with a Lucent Singularity, finales funkeln, Ignite, and a few autos. That might be enough to kill them.

Once you get blue buff, spam away. Buy sight ward to prevent any jungle ganks.

Blue buff is invaluable to Lux. You have long cooldowns, high mana costs, and low mana regen. Blue buff solves all these problems. Ask your jungler if you can have every blue after the first. It really helps a lot.

As Lux, you should be staying behind your team and spamming Lucent Singularity and Light Binding. Use your Finales Funkeln at the beginning of a fight to take out a large chunk of their hp. If the fight goes on long enough, your ultimate could come off CD and you get to use it again. Your Prismatic Barrier is very valuable. It can shield your entire team twice.

Oh yeah, remember to buy sight ward They help a lot. More vision means you know where the enemy team is so you know what they're up to. You can also catch a carry off guard and instagib them. Even if your support is already buying them, they're already gold deprived and could use some help. An extra ward or two doesn't hurt either.

Core Item Build is

Sorcerer's Shoes

Double Doran's Ring

Rabadon's Deathcap

Good luck, and have fun playing Lux!