Keep in mind that last hitting is a art, learning to control minion waves and your own mana takes time. Going with AS mastery over the AP one will cost you later, its better to just learn how to do it on your own with the standard builds that won't hinder you just to make CSing easier, you'll eventually get the hang of it, just takes practice. Like I said a couple of bot games just to simply practicing last hitting then ending and restarting after say 10 mins to keep practicing.
Different champs AA differently, Annie and Anivia have awful AA animations where as someone like Ahri and LB have better AA animations.
Mana managing is also difficult, early game try not to expend more then your 1 spell, Ahri Q, Ziggs Q, Xerath Q, the main wave clear spell. Also try to make it count, as Ahri try to Q the wave and tag the enemy laner as well. Also keep in mind by abusing your spell you will push in and make yourself vulnerable. If you notice your down to 1/3 mana, try to just chill and last hit it back up, try to never go oom cause you never know when you need to roam half way in the river to help your jungler or use your Charm as Ahri to help live a gank.
Always try to back with enough for your item, a Control Ward (unless your current one is still up you won't NEED to buy a new one) and 2-3 HP pots. HP pots are useful for most of lane until you realize they can 100-0 in a sec. When you back, try to push the wave straight into the enemy turret so the lane will reset before your back.
Backing for a 2nd Dorans ring will also help you with early game mana management, The passive stacks and you get a decent Mana regan/5, HP, and AP.
If you find that you have to back more then most people would, TP is also a good spell to bring with flash, I typically still run Ghost and Flash, but I sub Ghost out for whatever situational sum spell I need. Against the likes of Zed, Yas, Fizz and Talon Exhaust is most useful, if I'm the assassin and think I can bully lane I'll take Ignite and try to snowball. Against Syndra or Viegar barrier is good to protect you from there ults.
If your pushed up and see the opponent's pushed into your teammates tower early game, you can roam to try to help, get a kill or assist. Try to coordinate it with your jungler for a higher probable kill.
If you can, also try to hold your wave at the 30 yard line of mind lane, so the turret doesn't hit it but so you making the enemy kinda sit at the half way mark. Again wave management is hard, its a skill.
If you have any other random question, I'm down to try and help.