Erm, I have a guide for playing against Zoe, might help you a bit in what to do when playing as her, especialy against those who know how to play against her: https://boards.na.leagueoflegends.com/en/c/gameplay-balance/7TRVZ3Qr-playing-against-zoe
Best advice? Be creative. Don't get stuck int he rut of "every Q mus be the longest range possible" or "throw Q behind then infront" be creative in how you angle ur skillshots. Pay attention to the positioning of minions and how they'll move weave around them when trying to hit champs (in doing this it's best to know the general range and size of the splash dmg... it hits more perpendicularly than in a true circle. When possible, maximize ur passive. Q procs it twice, once per cast... so try to know the deadline u have before the recast time runs out.
You don't have to get to every single shard drop, though you should try to get each CS, weigh the costs of getting to the shard and the likelihood of u getting hit, then weigh the benefits. It's also a good idea to have a general idea of when you want to recall and which shard ur gonna pick up in that case as it can give u a ms buff to return to lane a bit faster. Keep track of the summoners dropped. And I mean over the entire map. They give a small zone of vision so can be extremely useful in keeping track of the jg/other laners.
E... try to land them lol.... remember it can go through walls, but will only go through one instance of terrain, and it's pop out range is gonna be based on how close u were to that wall. Keep in mind that the E can also be used in trap form to zone away from CS or block off choke points. Remember there is a windup before the sleep becomes activated.
R is where everything becomes even more decided based on how creative ur thinking, since it directly impacts her other spells by temporarily displacing you, this allows you to pop up unexpectedly over walls... or even when in lane quickly switch positions, thereby allowing you more opportunities. That being said, it's generally better to not use it to appear right infront of the enemy unless ur using it to land ur passive. People often base trajectories and predict them based on the champion's position, that's why R can be so devastating since it doesn't just allow for greater dmg from Q, but can let u land shots that wouldn't be possible without that quick position change.
In terms of creative uses for Q alone, u can throw it straight, throw it to the side then to the front, throw it behind then infront or even zig zag it through units to be even trickier. R emphasizes this.
In terms of Macrogame.. ward, as u should with any champ, they removed the vision granted from ult so it becomes even more important. Try to keep track of where ur E landed and whether or not it got triggered, Comet is the best rune to notify u on this, tho Aery also works (they both land before the sleep becomes activated so don't worry about sleep being popped).
My go-to build used to be Ludens/Seraphs, Lich, Sorc boots, Morello, Rabadons, Voidstaff... with a Morello substituting where needed.... Now, i almost never go lich, just cuz the Assassin meta really punished close combat fights, plus HP, penetration as well as Grievous wounds seems more important than flat extra dmg... So now my build is more often than not Ludens, Sorc, Morello, Zhonyas/Banshees, Rabadons and Voidstaff. Generally take Arcane Comet, Manaflow, Transcendence, Scorch, Future's Market and timewarp tonic... but that's me, ik Dark Harvest, Aery, Electrocute and even Unsealed spellbook are great on her... The precision treeline can also be excellent as a secondary, what with Coup de Grace and Triumph.
You can start with Dorans, Corrupting or, if ur feeling ballsy, Dark Seal.
LB u really don't want to mess with too much, especially if they're good, so try to get help from ur JG as often as possible, generally matchups are soemthing ur gonna have to learn by urself as it's largely based on ur playstyle. Sometimes it's best to hard push and roam, other times its best to stay in lane and try to keep it frozen and pray for a gank (kinda boring sometimes), always be on the lookout to be aggressive and land an E tho, as E- Passive Q passive can deal a huge amount of dmg, especially once paired with R. Standard combo once uve gotten ult is QRQ, a good Zoe will instead do QQR. It gives you slightly more range, and is less predictable... however depends on you having a good estimation of where the enemy will move unless paired with E.
Hopefully this helps, good luck, have fun playing Zoe. [slayer-jinx-wink]