Ahri is a high-burst-mage with a lot of movespeed so u should be always aware of ure capabilties.
In the early game: your top priorty is 2 push and keep ure lane warded her Q is lvl 1 pretty strong with tlords combined and make her a cheesy trader with alot dmg early on. So try keep that in mind and focus all on that.
Nr1 Q the wave at opprtunites where u can hit an enemy
Nr2 Never use ure w too farm, only under tower because of the mana cost
Nr3 Dont miss ure charm or use it carlessly, beacause it decides on u winning a trade or not
Nr4 Keep it calm! U have time on ure R, so Dont spam it! The first dash is too go in, the second is to get a little distance/doge burst and the third is too rengage with ure spell rotation up again too finish the deal
Nr5 Dont loose ure **** when u get ganked most of the time u have your ult up and maybe flash so use those, maybe u can even kill someone, when someone is low (who awaits a person to make a reengage when there are 2 ppl there ^^
Nr6 When u run, always Q behind u because as long as ure Q travels u have increased movementspeed and when u Q behind it logically has a longer travel time.

To the runes: The hybrid pen and attackspeed allow u 2 trade way harder in certain matchups and also help u farm a lot better, u can literally stick on a champ and auto attack him 2 death early on. Same goes for the Ad matchup nothing is more satisfying than getting that 1%crit on an enemy like yasou and it kills him.

Like I said early Dont loose ure ****! but what is the right spell rotation?
Its either: E,W,Q (R)
Or in an all in: W,R,E,Q,R,W,(Q),R

Pros: Ahri has a high burst and a lot of mobility which makes it hard too keep up with her, you can literally pick her in everything and ure True Dmg from ure Q makes it really hard too build against her.
Cons: Once u get CC on Ahri u are Dead in 90% of the cases, u lack the hp and resistances too tank a lot of damage

When u are new too Ahri u can probably never carry on ure on, because ure Goal in TF is too oneshot there backline and for this it will probabaly take u most ure spells and mobilty so its hard too make this 1v5 happen, but its possible.
Tipp: When ure team feeds but u are "fed" or strong get a sweeper wait in a brush and burst there squshies when they pass by.