Masteries and Runes should be self-explanatory, though ask if you don't understand. The early item progression is up for change, though I usually go that way so I can get early stacks to help my damage output but still get my tear as early as possible. DO NOT get RoA, you may like the benefits it brings, but it doesn't help your damage output like going straight for Soul stacks and and Archangel's does. For the final 2 items, I usually just stack 2 Rods and then upgrade both of them for the pure AP boost, but you may want to consider a Lich Bane or Rylai's instead, depending on game cirumctanches. Clarity helps early to even mid game with mana problems and flash is a necessity to at least try to escape ganks, though you should play cautiously anyway and only use it as a last resort, as your stun is the ultimate gank stopper.

Gameplay:

Early: Harass using Lucent Singularity and follow-up auto-attack to exploit her passive. If you have a good lane partner (preferably a nuker) then use your Q to stun and auto-attack while your partner nukes to death. Even if you don't get kills, the money is important at this point.

Mid-game: Don't be too aggressive at this point. Start getting stacks of Mejal by supporting your team with your Q and W and finishing them off with your ult. You might get a few complaints about Killstealing, but really, its a team game, so there really isn't such a thing. Even if your only getting assists, accumulating stacks and continuing to build AP is important here. Most people say the combo for Lux is to throw in e, stun, detonate e, and then ult, but that completely ignores your passive, and a lot of times they will be able to leave the blast radius of e before you stun them. Go q, auto for passive, e, detonate, then ult as they run away (igniting passive again). You'll be guaranteed to hit with e, and you don't need to ult while they're stunned, take advantage of its long range.

Late game: You should stay towards the middle to back of your team during team fights, anchoring them by stunning and slowing opponents, and lining up your shield to hit your team. You can even double shield people with a single throw, provided they're aren't getting nuked down too fast. Also, while spamming abilities is good, try to use your stun a little judiciously when things turn south in team fights, as you need to use it to stop enemy pushes in there tracks, taking advantage of its double stun effect. Also, keep your eyes open for fleeing enemies and snipe them with your ult. It takes practice, but after a while you can get good with shooting even through the fog. And don't be afraid to use it alot. At 20 stacks, you're cd will be ~30 seconds.