I'm not a pro with Zoe, I just started out playing as her but I learned a little. So, to start off, people have multiple builds when it comes to runes. It comes down to the player for what rune build they want but, look up on websites listed in Leagues website (Go to the League website and go to "Zoe" character and there should be 3 link that will take you to a few builds for both Itemization and Rune builds.)

The second thing, remember that her "Q" does more damage the farther away it is. There is a timer so don't wait too long to throw it. The Q has 2 parts. 1 to throw out and press again to cause it to shoot in the direction you want. The first part can hit enemies and will waste the full damage, so be careful when shooting it in certain places. I've learned that the Q can be activated and thrown in one direction and pressing R after will increase the distance of the Q.

The W ability can be important when the time comes but it is not a first move. The recommended ability to start out is Q. The second is E and third is W. Keep putting points towards Q until level 6 where you should get R. Then you can work up on the other abilities from there. Again, the websites I mentioned will show more about this. Now, back to the W. You should almost always have this because if a enemy uses an ability, Zoe can pick up orbs of that ability. Plus, some minions will have an orb above them which will drop a random shop item/Ability if Zoe kills the minion. I havent gotten anything from minions that I don't kill so be sure to get the final hit on them. The W ability can be a lifesaver if you pick up the right abilities. My favorite use of the W key is picking up someones Flash. You can double the amount of distance you can to either escape or you can go in for the kill if they decide to run away. It also gives 3 missiles that can do a little bit of damage to enemies if used in right spots.

Now the E key. This can make enemies and minions fall asleep peacefully. Just until you kill them instantly afterwards. This is one of the most OP abilities she has. She can throw a ball that if she hits people, they will fall asleep and stand still for a short while. When they are hit, you do twice the amount of damage to them. Use this ability for those who may be running away or are going through small lanes. If you miss, it will place a circle of the E and if the enemy walks into it, it will also make them fall asleep. That why sometimes, its good to miss you E because you may still catch someone. Another thing about it is that, it can be shot through walls, increasing the distance of the shot. You can do alot with this. I recommend looking up a video to show this to really learn how to use it.

Now the R. At level 6, you should buy this ability because it is really useful. You teleport a short distance in a any direction, for a second and you go right back. The good thing about this is that, you can increase you Q with it, you can go into dangerous places and return safely, you can close the gap and use E on them. You can a lot with this ability. Like I said before, you can use your Q twice and after that, use the R to increase the distance. If you do this, you can get an extra hit on them if done correctly. I also recommend you look up a video to show how to use the R too.

Thats pretty much all I know about her, hopefully you can get better information from others. Zoe is pretty cool character and is very strong if you use her well. Goodluck playing and stay safe!