Taking your "E" early and making combo shots on minions is key to your success, minding how much damage your combo does for best last hits. You don't have to worry about the enemy champions so much as playing careful and comboing down the minion waves, push where possible, use your shield ("W") for protection.

example:
You are laneing vs Kat, Kat is obviously going to tele behind you, slice you, and throw knives. to mitigate this damage besides the runes and build, you use your shield "W", drop a light sphere "E" on top of yourself(do not pop), snare "Q", combo luminescence (passive), pop sphere ("E" a second time), finish it off with your Shoop Da Whoop Laser "R", and push right on through) .

Lux, is very heavily skill shot based, in which target navigation prediction is a must and takes practice to learn, so keep trying it comes with time. the item build focuses on utility in team fighting and the ability to still do massive amounts of damage with her "E" and "R". I focus away from lich bane until later even though the combo benefit of it is amazing, this is because that shield is an absolute must have for getting out of stick situtations and adding AP will only make it absorb that much more incoming damage to keep you safe from jungle ganks etc. Please also take note that your shield is only as good as your ability to snare ("Q") the individual chasing you, especially if it is a ranged, snaring, or charging type charecter.