Lux is a champion who is heavily Mana dependent. She is also one of the weaker laning champion so keep that in mind.

Start

For the start, you can go on Boots + 2 Health Potion + 1 Mana Potion or as I do on Doran's Ring. It depends of the champ you have in front of you. If it's a champ who will harass you before your level 6 then buy potion, if not then go on doran ring to kill before your lvl 6 with, or without your jungler.

You have to focus on farm (you can use your Lucent Singularity to apply Illumination on all the wave), but also to harass with your E, and then with an auto-atack. When your opponnent is half-life, you can try Light Binding + auto-atack + Lucent Singularity + auto-atack + Ignite and it will give you quite always a kill !

Mid Game

After your level 6, you have to start to roam. Your high range skill shots will give you easly one or too kills on bot lane.

Late Game

In late game, you have to use more your auto atack after a spell than your spell ! They will dealt always more, and you will be able to down easly all the ennemy team, even the tanks !

Team fights

In team fights, you must be behind all your team ! Try before to engage to harass, if you are level 16 and that you have Morellonomicon, spam your Final Spark. Try to use your Lucent Singularity before your Final Spark for the bonus damage from Illumination.
Just becarefull not to be catch. Even if all your team die, you will be able to protect and take the last kill if you survive.

On start you can choose between 2 starts. I prefer the Doran's Ring one with this champ, because I can kill easly before my lvl 6 with it. But if you are against an hard harass champ, and that you are not too sure about yourself, then go on the second more classic one.

Lux Spells have huge cooldowns on start. That's why I buy Athene's Unholy Grail and Morellonomicon. With this 2 items, you will have all your spells at their lowest cooldown. When you will be lvl 16, you will get your Final Spark all 24 seconds for about 1K damage !
The Lich Bane will makes you viable in long game, and you will be one of the more powerfull AP champ in long game.