Hi and welcome to my first ever guide on Mobafire. I made this build to see what Lux would be like if she reached the 40% cooldown reduction cap, so if you want a high AP build, there are many out there you can choose from. But none of them will truly give you a 24sec cooldown laser (at lvl 16 of course), and THAT is pretty neat imo.

Greater Mark of Insight - 9 - Magic Penetration
Greater Seal of Clarity - 9 - Mana Per Second
Greater Glyph of Focus - 9 - Ability Cooldown 5,85%
Greater Quintessence of Fortitude - 3 - Health

Pretty basic for a mage, with of course the cd reduction runes (flat) to reach the cap, some mana regen (although I yet have to see the latest patch release notes to see how much they nerfed those) for early game, some magic pen (because it's better than AP!) and of course, because you are so squishy, health per level quintessences.

For masteries, standard 9/0/21 build. With the points corresponding to your summoner spells or whatever you feel like, just make sure you max Sorcery and Intelligence.

Flash: Amazing to get away from any dangerous situations, and hit your passive to get the kill in early game or position for your Light Binding. The fact that this build has no survivability is also a decisive factor in the choice of this spell.

Clarity: I like this spell because its a real killer. Yes, I do mean killer. Because when your enemy see's you low on mana, that's when he gets more aggressive. Also, if you have harassed him enough before level 6, you're completely oom, and that level up only gives you enough for one spell. So what you want to do is send him a nice Light Binding, Shield yourself, Clarity, then Lucent Singularity, then BOOM, he died and he never saw it coming.

These are my personnal choices, however, some other good summoner spells for Lux are:
Clairvoyance: To see where you're ulting!
Exhaust: Good spell, just to slow them some more, and lower their MR with the mastery, or to escape when things aren't going your way.
Ignite: If you absolutely want to kill your mid lane enemy before level 6. Becomes a bit useless late game though.

I used to start out with a Doran's Ring and health pot, but due to the recent changes, I think the pots are still really important, and so is the mana regen, so now I start with a Meki Pendant and two health pots. The logic behind it is the following: when you could buy DR and a pot, you had 100 more hp, and a 200hp heal with some mana regen and, of course 15 AP. The loss of AP is the only thing I regret, but otherwise the 100 hp loss is widely compensated by that extra 200hp heal, AND you now have more mana regen, so you can actually stay in lane a bit longer, and may even ditch your clarity spell (although I don't recommend it because Lux is a mana hungry champion). After that, when I hit 1100 gold, I blue pill and get Sorcerer's Shoes. However, for all you low levels out there who dont actually have all your runes and masteries, and therefore are missing that precious 15% cooldown reduction (it really is 14,85% but hey...) you should get Ionian Boots of Lucidity. After that, grab a Fiendish Codex, a Stinger and build your Nashor's Tooth. This item is the whole reason of this build, because, getting that 25% cooldown reduction is just a must since you just reached the 40% cap, still got some AP, and are starting to get enough mana regen to shield yourself when oom. Some of you might argue that Frozen Heart gives the same cooldown reduction. Now, if you are facing a lot of melee auto-attackers (like that pesky Xin Zhao)by all means, take it, but the reason I choose Nashor over Frozen Heart, is the 50% AS and 55AP. No need to explain the advantages of the AP, but the AS? might you ask. Well, there is a simple explanation for that: your passive; the thing is, if you manage to land an auto-attack after every spell (see the skill sequence part for more explanation), at level 18, that's 109+190=299dmg (before MR and Armor), twice or even three times if your ult didnt kill him, so who is going to spit on 600-900 dmg? Didnt think so.
Next, it's all basic mage build; get Tear of the Goddess, then start building your Rabadon's Deathcap, starting by the Needlessly Large Rod. After Rabadon, I usually finish Archangel's Staff before going to Mejai's Soulstealer.
You might notice I didnt put in a 6th item, and thus for two reasons:
1. The game is over, GG.
2. It is highly situational, ie. if the other team is stacking MR, i get Void staff, if they are CCing me every chance they get and are shredding me to bits, I take a defensive item corresponding to their most fed carry's type.
Ideas for 6th Item:
Rabadon's Deathcap, for that extra 155 AP!
Void Staff
Rylai's Crystal Scepter
Thornmail (for AD champs)
Zhonya's Hourglass (100 AP is nice, and gives you armor + invincibility for these crazy ults)
Banshee's Veil (MR + Spell immunity)
And the list goes on...

For the skill sequence, the level where you get Prismatic Barrier isn't very important, although I do advise you to get it before lvl 6, because you might get ganked, and, well it's a shield, so even if it doesn't absorb that much damage, it can actually save your life; I can't count how many times I read "OMG LUX 3hp!!!!!!!!! AAAARG". Also, as I used to main Karthus, a good Lux prevented me from getting a quadrakill with that, so think about it.
As a first spell, when mid lane, I'll go with Lucent Singularity (LS), then Light Binding (LB) to start harassing immediately and last hitting creeps (dont forget your passive). For side lanes, ask your laning partner what he likes most. Melees will usually like you to start with LB so they can have 2-3 hits on the enemy before he even starts to retreat. Still, if you want, you can always start with LS, and use it to scout the enemy bush, but if they're not here, well, you've just wasted 70mana and that's not cool. My advice is, take both at level 1 and 2 in the order you prefer, then get your shield, and then, max out LS. Why? because when you are laning, LS is much easier to place than LB, since it doesn't get blocked by those damn minions and your enemy will generally stay behind his caster creeps, and from behind yours, you can send your LS right behind him. After throwing it from the safety of your caster creep line, right click on him, then detonate before your projectile hits the enemy, and retreat to lose enemy creep agro. Sometimes, go to the side to get a clear shot for your LB, and quickly do LB, then LS, right click the enemy, wait for the first passive to blow up, then detonate LS and hit him again before retreating. Keep this up until you get your Finales Funkeln (your Laser). Also, make sure if he is blue-pilling, to send an LS (by staying out of range of tower of course) to disrupt him, and keep him in lane until you can Laserise his face!

So, basically, what you can see from the skill sequence, your priorities are as follow:
Finales Funkeln > Lucent Singularity> Light Binding> Prismatic Barrier

This build isn't done yet and still might need some tweaks, so feel free to comment (constructively please). Also, I'm still not sure about the skill sequence, as Prismatic Barrier is still pretty useful.
Oh, and keep in mind, I don't rely on Mejai's 15% cooldown reduction, but on the runes and masteries instead, unlike that other build I saw.

Changed

Clarity for Teleport after an effective try out and Domainoft's advice.
Swapped Mejai's Soulstealer with Tear of the Goddess for an earlier and wider mana pool. Note that I still finish Rabadon's Deathcap before Archangel's Staff as the staff's AP is much lower than the deathcap, and the only bonus stat is mana and mana regen, which we're getting from the Tear.
Changed the skill sequence and masteries according to summoner spell changes.

Thank you Domainoft for your constructive comment!