Okay, I've been playing for only about a week, but I've found that the best way to be an asset to my team is to pick Tristana and buff her for attack speed and damage. I focus on maxing her Q and treat her like a kind of damage over time gal.
Are there any other ranged champions that are more suited to a damage over time style?
Malzahar.
It's generally better to max E first since it basically amplifies your autos and gives you a ton of damage to towers, then Q second.
You've got the right build for her, lots of AD, crit, and attack speed so you can abuse her long attack range in the late game with more attack speed from Q.
Most marksmen focus heavily on their sustained damage. Their role in the game is to provide constant, dependable damage to their team to help focus down tanks and take objectives. Caitlyn uses a long auto range and her traps to make it really easy to siege towers. Ashe is kinda similar with her rampant slows. Jinx and Kog'maw can have very long auto attack ranges for certain periods of time (Kog'maw's only lasts a limited time and has a cooldown, Jinx has to spend mana and give up attack speed) Vayne has a mediocre attack range but makes up for it with mobility and bursts of %hp true damage every 3 consecutive hits. Most other marksmen have decent sustained damage but their kits focus more heavily either on burst from their abilities (Varus, Corki, Ezreal) or powerful but brief windows of sustained damage through their ults (Miss Fortune, Twitch)
Outside of the pure marksmen there are a few mages that have low cooldown abilities that allow them to provide similar levels of dps (Orianna, Ryze, Cassiopeia, and Karthus); some ranged auto attackers that enhance their autos with AP on-hit abilities (Azir, Kayle, Teemo); the light fighters who maximize their auto attacks but need to get in melee range to deal their damage and have to be careful with their engages and defensive abilities to keep from getting blown up (Master Yi, Tryndamere, Fiora, Jax, Yasuo); damage-focused tanks that can get into the fight and just deal a bunch of indiscriminate damage while being hard to kill (Amumu, Zac, Skarner); and diving fighters with very strong access to the back line and on-hit damage that helps them take down targets with hopefully just enough tankiness to succeed (Shyvana, Nocturne, Aatrox, Irelia, Udyr)
