As Lux, your greatest advantage is your unrivaled range. You have significant burst damage and some utility, but your range is your biggest friend. Not just Finales Funkeln either; all of your offensive spells have a massive range advantage over any champion you will lane against. As mid (which you should always be) it is easy enough to secure a one level lead VERY early game.

Terms and abbreviations used in this guide:
Autoattack (AA) - your champion's normal physical attacks
AP - Ability Power
AD - Attack Damage
CS - Creep Score
CDR - Cool Down Reduction
LOS - Line Of Sight

Pros:
1. RANGE. You can easily out range any other champion with any of your offensive skills. Even your autoattacks have comparable range. This is your biggest asset especially early game. Your ability to out harass will give you an easy edge.
2. Burst damage output. Lux's ability to dish out massive damage to a single target is almost unrivaled. None of her abilities do damage over time, so your hits are guaranteed to do full damage if you can land them.
3. Survivability. While Lux is fairly squishy, she is slightly less so than other AP carries. Her snare can stop two champions unlike Morgana and does not require any combo to pull of unlike Xerath. Plus her E can be used as a slow in a pinch, adding to the escapability factor.
4. Free wards. Use her E to check bushes and you will never be caught off guard.

Cons:
1. Lux is an AP carry, and therefore squishy. Proper flanking can nearly negate the utility of her snare and once her snare is on cooldown she's almost a free kill.
2. More relating to the best strategy for Lux, her general detachment from teamfights is a double edged sword. While she can generally avoid damage this way, it can make her easy to pick off if you aren't careful.

For your rune and mastery pages you want to focus almost completely on AP. The magic pen. reds help in getting around magic resist items mid game, which your opponent will most likely build. Mana regen yellows help keep you from running OOM, especially early game. AP/Lv blues are there because they turn around fairly early game over the straight AP runes (at level 6) as opposed to the AP/Lv quints, which don't turn around until level 12. As an option, you may experiment with AP/Lv yellows, but they are significantly less potent than the blues and you will probably have mana troubles later.

Masteries are very straightforward, AP, magic pen, a little CDR. Not much explanation needed there, just damage. Also Summoner's Insight is EXTREMELY important because it improves Clarity by 20%. That necessity is explained later.

Yep. Here is why: Sheen costs 1260 to build, and adds bonus damage equal to your AD to the next attack you do after using a skill. Lich bane costs 3470 total to build, and does the same thing only it adds damage equal to your AP, as physical damage. Did I mention that this bonus damage is physical damage? Your opponent mid will be building magic resist items, and this will wind up being your primary source of damage.
For reference, Rabadon's Deathcap costs 3600 to build.
Let's do some math. Since you should (will) be autoattacking to crack your passive on people for the bonus damage, we will assume that you successfully land an autoattack after each ability you let go. Let us assume level 12, which is approximately when you should be able to buy either of these items (it may be slightly later for Rabadon's, but I digress).

(Feel free to correct any math errors I might have made, I could have missed something.)

With Rabadon's Deathcap, you have 654 AP. You can already see where this is going:

Light Binding: 210(+458) (Autoattack: 73+120 from passive)
Lucent Singularity: 240(+392) (Autoattack)
Finales Funkeln: 400(+490) (Autoattack if you're lucky)

= 2769.

With Lich Bane, you have 443 AP at level 12, including your runes and masteries. Here is the kill combo:

Light Binding: 210(+310) (Autoattack: 73+120 from passive+443 from Lich Bane)
Lucent Singularity: 240(+266) (Autoattack)
Finales Funkeln: 400(+332) (Autoattack if you're lucky)

= 3666 damage, of which 1548 is physical. Ouch.

Of course you can add Rabadon's after that for even more pain...

The reason this works so well is because you will already be trying to crack your passive on your opponent, so Lich Bane's damage is destined to happen anyway. It also adds a small element of surprise, since players will not expect your AA's to do over 500 damage each.

You will start the game with half boots and mana pots. You might opt for an Amplifying Tome to speed your acquisition of Sheen, but the movement speed boost has saved me enough times that I usually keep the boots. Start with your E (unless your team calls blitz, then start Q) and then take the other one level 2. Once you have them both focus on your Q and only take your W when the system forces you to. It is basically useless.

Use your E to harass the outside edge of the caster minions to harass. You will either deny CS, or get early damage. Also make sure to capitalize on your passive as much as you can. Your passive will crack for 10x your level which starts to add up pretty quick at level two. Also use your E to secure last hits. Lux is one of the best farmers in the game.

You really shouldn't NEED to B until level 8, but if you have a good enough lead by level 6 and enough gold to buy Sheen outright (1260) then go for it. The reason I take Sheen (into Lich Bane) instead of rushing Rabadon's is becuase a level 6 Sheen allows you to take one extra guaranteed CS per wave. Normally at level 6, Lux can last hit the caster minions after letting go of an E. Sheen gives enough bonus damage to kill a melee minion in addition to the three caster minions. You use your mana potions during this time, and NOT Clarity.

Once you have Sheen, you are ready to secure your first kill. Harass with your E until you are getting low on mana, then let your opponent push close enough to pull your entire combo, then Clarity and go for it. This is when you abuse Heal. Heal taunting is hilarious with Lux because it can let you get through your combo twice in a single encounter. This is why I take Heal over Flash. Even if you don't actually get the kill, you should force them to B. Push your lane into the tower (take it if it's close) and then start ganking hard. By this time you are easily 3-4 levels up on the lanes, and you can do some pretty serious damage really fast. coordinate ganks with your jungler if you have one to the double lane for a 4v2 and use your ult to secure any runners. An effective strategy during this gank is sit in the jungle behind the enemy tower and wait for someone to run back up the lane from the 3v2 gank, then pick up the easy kill while they're running.

Starting at around level 11 or so you should be able to get blue buff without much trouble, and you should any time it's safe and available. Make sure to start warding Baron also.

By the time you get past Level 12 you will be ganking more than laning. When initiating a teamfight, make your team wait for you and you hold point. As Lux it is best if you take point for a couple reasons:
1. Gain LOS from your E and check bushes.
2. Teamfights that open with a snare are quick, easy victories.

Once you get contact, blow your combo (minus ult) on a single target and GET OUT. As Lux you want to use your range as a defense and keep yourself out of the fray. Throw in your E to help the damage along, Q to stop runners and if someone does manage to get away use your ult to pick up the extra kill.
CONTRARY TO POPULAR BELIEF: Even if you can't land a good kill, an assist is always worth it. If your attempts result in a stolen kill don't worry about it too much, because a KS adds team gold by adding an assist. Don't be a ****, and if you already have an assist on the kill just leave it at that, but if someone already has a double kill in the teamfight don't be afraid to snatch up a spare with your AOE or something. You are the carry, and you need to get Rabadon's on top of your Lich Bane as soon as possible. Once you have Rabadon's though don't worry about gold too much. Your team might cry about it a little, but the faster you get Rabadon's the faster your team wins. Once you have Lich Bane + Rabadon's Deathcap you have easily the most damage potential out of any champion in the game.

Another tip: while Lux's shield is basically useless, don't forget about it during ganks. It will rarely help any, but if you can hit enough allies with it it's worth the little mana it costs. A well placed staff throw can save a teammate who has been ignited, etc.

1. Use your range to harass for an early lead and keep yourself out of immediate danger.
2. Be wary of Flash, or abilities like Katarina's Shunpo or Kassadin's ultimate that can close in on you when your snare is on cooldown. If you are playing against a champ that has such an ability, count their cooldowns and play very defensively. These are Lux's biggest weakness. Heal taunt them into a turret dive and snare them if you get the chance.
3. Do NOT face check bushes. Your E grants line-of-sight, so use it. It's worth the mana.
4. Don't get caught alone travelling between lanes, or ganked off guard. Ward the river, and use your E. If you get flanked you're done.