By now hopefully I've explained why pushing midlane is so important, but how do we actually push waves correctly?
The obvious step 1 is making sure you're hitting as many minions as possible. This sounds obvious, but you would not believe the amount of times I've seen students not do this in coaching sessions.
After you've used your spell on the wave to push, comes the most important part of pushing. Walking away. Think about it, you have used your spell on the wave to push, what is the enemy laner going to do? He is going to try to trade with you because you have a spell on cooldown. So, the way to get around that is by walking away, waiting for your spell to come back up and then push out the wave.
When you get later into the game this will not matter because you reach your waveclear breakpoints. Which is the point at which you can oneshot the wave in one spell rotation. Once you reach this point on Ahri, that's where the game starts, now you can oneshot every single wave and then roam.
Another thing that matters when pushing the wave early game is hitting the minions. You should ALWAYS be hitting the minions to push, because you push quicker, so your roams will be that bit faster.
By pushing the wave you force the enemy laner to choose between using their spells on the wave — which then opens up a trade window for you when you get your spells back up — OR try to trade with you to punish your cooldowns, but we already covered how to avoid that earlier.
The key takeaways are: learn how to push efficiently, learn your waveclear breakpoints, force yourself to always auto attack minions, and remember to respect your own cooldowns when you use spells on the wave.
This is the primary way I beat enemy laners without ever having to interact with them.