Hey all,
I started playing Lucian during his first free week and wanted to find a guide to follow for the first game, since I was unfamiliar with this new adc. Yet there were no guides above 78% or so when I checked MobaFire. I sallied forth on my own, making my own build as I went along, and have found plenty of success so far, getting a pentakill within my first few games, so I decided to publish my own guide. Enjoy!

First things first: Lucian is an adc, at least so far as this guide builds. As such, I recommend the usual adc spells, Flash and Barrier. Lucian's increased movement speed after building the PD and perhaps a TriForce combined with his dash give him very high mobility; Flash works very well with this for quick escapes.
Barrier is generally a good choice for the other summoner spell; Cleanse could also be used, especially against high-CC team comps, but is not always necessary as his dash removes slows from him, which can save him if caught by, say, a Sona slow, Thresh's ult, or the slowing effect of Leona's ult.
Ghost could also complement Flash well. Like Ryze, with his movement speed-increasing ult, Lucian, with his dash, could have amazing engage/disengage with both, although it would sacrifice offensive/defensive capabilities.
Ignite is also a viable option, especially combined with Lucian's chasing abilities, although I typically don't use it and usually have no trouble finishing off enemies without it.

Again, Lucian is an adc. 21/9/0 Masteries are a suggestion, as are mainly offensive runes with flat armor and MR per level thrown in. A bit of defense is useful; however, adding utility to masteries at the cost of defensive capabilities is also an option, so long as you're willing to deal with a squishier champ. As far as offensive runes, I suggest a combination of AD and armor pen, as both are important to Lucian. You could also experiment with only AD or only armor pen, although Lucian's damage for different targets will be less balanced that way. You could also use movement speed quints to increase mobility, although I have not had much trouble with having enough mobility on Lucian.

The key to Lucian when using this guide, I've found, is to understand that he only becomes really dominant later in the game. I see Lucian as somewhat similar to Vayne with this guide: especially at early levels, he tends to rely heavily on his auto attacks to farm and fight. This is largely due to his extremely mana-hungry nature at early levels. Thus, the typical Doran's Blade start is suggested. I would then suggest the Berserker's Boots, rather than a damage or life steal item, to start. This is for several reasons: if there is trouble in lane, this item can be bought early for cheap to help maintain control; it gives Lucian both increased AS to help farm and heal with the Doran's Blade and early mobility, good for trading, chasing, disengaging, and avoiding ganks. His ult also stacks with AS, so he can get more early/mid game damage this way. To complement this, rush a BOTRK afterward. This early combo doesn't give Lucian lots of burst damage, but lets him chase more effectively and keep doing consistent damage with the slow and % health damage from the BOTRK.
Come mid game, though, it's time for Lucian to really start doing some damage. Getting the standard 2-item crit combo gives Lucian more overall damage and especially empowers his auto attacks. Your combo, explained in the next section, will do the damages.
End game is where I suggest creativity. If you're doing well, and especially if the enemy team still has little armor by this point, a TriForce followed by a Last Whisper can really be devastating. If they're heavy on the armor, combining a Last Whisper with a Black Cleaver can help you melt your targets. For general defense, a GA may be a good choice, although something like a Frozen Mallet or Mercurial Scimitar can offer a good mix of offensive and defensive capability.
As for enchantments, Furor gives excellent chasing potential, especially when combined with Lucian's kit, although Homeguard and even Alacrity are workable choices.

Lucian's skills are the biggest part of what makes him so useful. Here are some notes on each.

Lightslinger: Every time you use a skill, your next attack within a few seconds adds another, half-strength basic attack. They even apply on-hit effects, so Lucian can do amazing damage by alternating skills with double hits continuously. Going between high-damage skills and double-hitting, life stealing, % health damage-dealing, armor penetrating, even double-critting basic attacks can easily equal GG.

Piercing Light: Hit an enemy and any others near enough behind and even slightly to the side of the target. Useful for poking and farming, both by targeting the desired victim or an enemy between you and your victim. More effective for laning than Ardent Blaze, I suggesting starting with and maxing this first.

Ardent Blaze: Fire a projectile a certain distance; after hitting an enemy or reaching the end of its range, it detonates, doing damage to the enemy (if hit) and any nearby enemies in one of four directions. Also reveals and marks enemies hit; hitting a marked enemy gives Lucian a temporary movement speed bonus. Useful for fighting, but especially for checking bushes and, with the movement speed bonus, chasing. I suggest getting this third and maxing it last unless you really want the extra damage earlier.

Relentless Pursuit: Dashes a short distance, removing slows. Very useful for escaping slows, getting over little walls, etc. I suggest getting this second and leveling up after Piercing Light, trading off between points in Relentless Pursuit and Ardent Blaze. Maxing AB first will potentially increase damage output, but leveling RP lowers it's CD and mana cost, which helps in mid game for mobility and mana-saving purposes.

The Culling: Fire a barrage of bolts in one direction for three seconds. Can use RP while active, and if The Culling kills an enemy, RP's cooldown is eliminated. This is further reason to level RP early, so that it can be chained with The Culling kills without running so low on mana. The Culling stacks with AS, making the BOTRK/PD combo useful; however, it seems to cap at 28 shots, so adding a TriForce may not increase the number of shots. Also does 4x damage to minions, so it can be used as waveclear in a pinch. Max ASAP.

In lane, Lucian is not the most powerful champ. This build can be extraordinarily powerful and monstrously efficient, but it takes time to build up. Avoid overusing skills and try to keep your distance from champions with stuns, e.g. Leona. If you have a support who can heal, trust in their sustain to keep you alive so long as you're cautious. A q-double attack can be a good trading mechanism, especially if you have a support who can stun. A well-timed RP can save you from almost any skillshot combo, e.g. a Lux snare. As you build up the first two items, farming and trading will be easier.

As Lucian moves toward his IE and PD, his DPS increases dramatically. He is still squishy, however, and before getting 3 to 5 items built he can have trouble killing enemies, especially with little help, unless he has enough time to steadily pump out damage. Buying the PD and upgrading RP 2 or 3 times will help keep Lucian mobile and allow for timely engages/disengages. For best results, chain skills with basic attacks, e.g. RP in + basic attack + AB + basic attack + ... By this point you've closed the gap and can keep up with the move-speed bonus from AB. Be careful when ulting, as some champs can easily avoid or block the damage (as can minions).

Lucian is a beast in the late game. As he approaches full build, he does tons of damage and can burst enemies to death quickly. If building full damage, take care not to overextend, but understand that Lucian also has great potential for disengaging. His team fight potential also increases with both his ult and AB having the potential to hit multiple enemies. This is where Lucian becomes legendary.

Lucian takes some time to get off the ground, and laning phase can be a bit harsh. However, surviving to late game, Lucian gains both insane burst damage and continual DPS, and may lead to tons of damage, legendary status, and even pentakills. Good luck and have fun!