This is what makes Lucian a unique champion. This enables him to do more burst from a Trinity's force, it also makes him favor CDR items and enables him to optimally use items like Black Cleaver.
Q: Piercing Light
This will be your bread and butter, you will always max this first. It has a huge AD ratio and will enable you some additional distance to poke through creep waves with.
: If an enemy is escaping to their turret and there's creep nearby, try to lineup to Piercing Light a creep just in front of them to gain extra range.
W: Ardent Blaze
This is your longest-range poke which is purely a skill-shot. You will max this second. This can be excellent for last-hitting a fleeing / low health enemy who tries to juke, or it can be used to poke long-range (for turret-sieges)
: This will hit the first mob it comes in contact with and fires in a star. Familiarizing yourself with the path of the star can maximize its effectiveness and at times, shooting into a mob so it will explode at the right place can be better than attempting to blindly hit an enemy who may have ran to the fog of war or flashed over a wall.
: Use this prior to "The Culling" to make it impossible for people to outrun your ultimate.
E: Relentless Pursuit
This is another anti-kite and/or escape. You can use this like Renekton's "dice" to go through short walls for escapes or you can save it for a slow or snare, as it will remove any current slows / snares that are on you.
: Depending on your opponent's composition, you may want to either use this offensively or defensively (if they have a lot of snares or high mobility opponents).
R: The Culling
This is a fairly good burst ultimate that receives benefit from both AP and AD. You can move forward, backward or strafe left and right while casting it but can-not change the angle of the shot. Often times, shooting it parallel to the direction of the lane is optimal and just strafing to continue hitting.
: the range on this is fairly long, and it makes an excellent anti-chase deterrent. While it does provide good burst, I feel running through your Q -> W -> Q -> E combo once you've maxed CDR will yield better burst damage than R. Save this for when it's most valuable.
Additionally, while ulting, you can combine ghostblade with it to maximize the benefit from it.
: Use "Ardent Blaze" before to make it impossible for people to outrun your ultimate.
Your typical combo with Lucian will be Q -> W -> Q -> E (repeat). This will allow you to basically spam after waiting for your passive proc from each.