My first priority on Lux is mana regeneration and cooldown reduction.
My first priority is to be able to support my team, scaling my damage later.
Light Binding, Prismatic Barrier, and Lucent Singularity are excellent spells for securing a gank or rescuing yourself or a teammate from trouble.
In my opinion you're better off having a weak snare/slow to catch an opponent with than a strong one that's fired too late.

The two philosophers stones are arguably a bit of a waste as of patch 1.0.0.120
I have not yet decided if/how I'll alter my early build order.
Main possibilities are: change nothing / rush a Reverie / buy a philosophers stone and a tear of the goddess (which is what I used to do).

The goals remain the same:
Priority 1: mana regeneration and cooldown reduction
Priority 2: ability power

With the runes and masteries I use for Lux, having 15% cooldown reduction from items gets her close to the maximum cooldown reduction, having a less than thirty second cooldown on Finales Funkeln at level 18. This allows her to often use it early in a teamfight and then afterwards on a fleeing enemy.

Throw out a snare.

If it hits your opponent (and your spells are ready), you have two good combos to follow up with:

Snare to slow with two procs of illumination:
As soon as the snare hits, throw out a Lucent Singularity (E). Commence autoattacking.
Detonate the Lucent Singularity immediately after your first autoattack hits (proccing Illumination).
If you are able to auto-attack again, illumination will proc a second time.

Nuke:
As soon as the snare hits, throw out Lucent Singularity (E) and Finales Funkeln (R) as quickly as possible (hold shift, point at your target, and hit E then R quickly).
Mash E to detonate Lucent Singularity as soon as possible.
Follow up with an auto-attack if you are able.

Note that if your target has merc treads you cannot guarantee the nuke combo if you wait to see if the snare hits (I think. I have to play with 200 ping.) You can throw out your spells if you anticipate that a snare will hit, but your targets will unexpectedly dodge at the last possible moment some of the time, therefore remember in this case the nuke is not a true combo and has a risk vs reward factor.

The most efficient way to farm a minion wave when you are able to free-farm is:
Wait until the opposing minion waves collide.
Position your Lucent Singularity so that it will hit all 6 minions in the wave.
Throw an auto attack at one of the caster minions.
Detonate the Lucent Singularity just before your auto attack hits, so that it procs illumination.
This kills the caster minion (once you are above level something)
From here you should be able to last hit every single minion in the wave with illumination, just pay close attention to the damage being recieved by each minion.

There will be times when it is appropriate to push a lane with your ultimate.
These are most notably when multiple enemy champions want to attack your tower but still need a minion wave to do so (and your teammates are absent or lack ideal damage output).
(Note: the first level of Finales Funkeln costs less mana and does more damage than rank 5 Lucent Singularity. Although the cooldown at this point will be very long, it can be your best option if you are very low on mana and trying to prevent tower damage.)

It can also be very effective mid to late game to evaporate an enemy minion wave instantly without even appearing on their minimap. If this gets your minions to an enemy turret, you can be long gone by the time someone's rushed over to fix it.

Obviously, when you have enough AP to one-shot the entire wave this needs no explanation.
Before that (and when you have enough damage that your ultimate does one-shot the casters), here's how to do it:
Make sure you hit the enemy minions while they are still in a line.
Throw out your E+R the same as the nuke combo above, making sure the E hits the melee minions.

In the few levels where you have your ultimate but can't one-shot the casters with it, simply hit the entire wave and then carefully time/position Lucent Singularity so that it kills all six minions.

Killing the wraiths and wolves is simple.
Killing other jungle creeps before you have a lot of AP is not worth your time.
If you want to take the blue buff, I recommend getting a teammate to help you.

Besides the bleeding obvious, like hit as many allies/enemies with your abilities as possible in a teamfight and try not to be in range of anything painful.

So yeah, that's how I play Lux, do what you like with it and good luck.

Now tell me how to Cait >=d

P.S. have a video Play Video