How do I improve my early game

Stop being aggressive and try to get as much cs as possible. Ward your sides to look for jungler

FARM A very good trick I use is NOT auto the enemy and only focus on last hitting as much as possible. So say your enemy hits you when you get some CS, well then all your minions will attack them... it will do almost as much damage as if you had hit them back. Then they have wasted possible CS making an even trade.

DON'T CHASE OK you got the enemy to 100hp but you've used all your abilities. Instead of chasing to the turret and possibly dying to a new wave of minions, the turret, or a gank from the jungler, return to your wave and get CS. This forces the enemy to stay in the lane until your abilities go off CD granting you a much easier kill scenario or they have to back making you have more XP and gold than them. It's a win win. Not chasing is always much better than chasing.

GANKED BY CAMPER Jungler camping your lane? THIS IS A GOOD THING! This means that your jungler is free to grab towers, dragon, kills... whatever and they can't cause the jungler is waiting in the bush to kill you. If you play smart you can keep your lane even and the other lanes can snowball. Ward the bush that jungler uses to come in and let your enemy push the lane up to your turret. Farm just before the turret on on the turret. Miss a couple CS to let them push. Sometime you can even get a kill 1v2 from a dive.

Any particular lanes/champs?

Watch the basic mechanics videos from here.

Do these exercises to learn to farm better.

What role do you play?

a good start would be telling what champs you play what lane than describing your playstyle your build your runes so that people could give actual comments that will help you

if you just want some general tips heres 1 pick champions with early game power