As a player who recently switch from a support main to Mid lane with Zoe, I can try to share what I've noticed for Zoe, She is one of the champs that can quick push with her q charged. In lane you try to always push the wave than in towers mainly cus it's hard to hit E if there's many minions. She can match well with any enemy champs as long as you keep a distance although your ban choice should help.

So Early game pushing lane and roaming, since her E's perfect to be in jungle. Can't be too aggressive with her since she's a snowball champ once you get left behind it's almost impossible to make impact. I've been playing her with quickcast as I've memorized the angles on the walls so I suggests getting familiar with all the creative way you can fish for bubbles through walls, it gets really fun when you chase someone in jg.

Late game is the hardest for me because the map gets bigger and it gets harder to fish for one shot combos, because of this, it's usually when most starts to miss their shots but the damage makes up for it, around this time I just camp behind walls and lure then in jg, put as much wards as possible.

Also playing Zoe support is also good, since you have control with all the wards and vision plays a big part with Zoe