PROS:

• Very Fun
• High Burst
• Can 1v2 Very Easily
• Adorable
• Highly Adaptable Thanks to Spell Thief

CONS:

• Extremely Difficult to Master
• Immobile
• Predictable When Using Portal Jump
• Squishy
  
  Playing Zoe is a truly magical experience. Your job is to get in there and cause as much confusion and mayhem as possible while you bounce Paddle Star and Sleepy Trouble Bubble around the lane. Your goal is to burst down or harass as many people as possible while remaining out of reach. Let's get floopy! That's a word, right?

Ability Insight:

• Instant Glitter: Any auto-attack modified by More Sparkles! becomes an instant-contact attack, similar to Vel'Koz's or Azir's auto-attacks. This means that More Sparkles! auto-attacks pass through abilities such as Yasuo's Wind Wall.
• Unlimited Glitter: More Sparkles! comes into effect once Zoe casts any ability, including both casts of Paddle Star, using Portal Jump, or any iteration of Spell Thief.

Ability Tips:
Use More Sparkles! when trading with your lane opponent early game. Since it is instant-contact, there is absolutely no way for them to dodge it. More Sparkles! is also Zoe's primary method of last-hitting minions, since it provides a very substantial chunk of damage (and Paddle Star will not one-shot minions until mid-to-late game).
It is also possible to finish off an opponent by using Portal Jump and smacking them with a quick auto-attack from More Sparkles! when they try to escape.
This passive has no cooldown, so use it whenever you can! It also works on turrets, so mid-to-late game, Zoe can take down turrets with ease as long as she threads More Sparkles! in between her abilities. Ability Insight:

• Trick Shot: Paddle Star gains damage the longer it travels in a straight line after the initial cast, meaning that you want to maximize your distance in order to get the most bang for your buck.
• Sparkle Splash: Paddle Star detonates in a small area-of-effect, dealing 80% of it's total damage to surrounding enemies.
• First Star on the Right: Paddle Star's cast range follows Zoe's current position, so by using Flash, Portal Jump, or other quick movement abilities, you can extend the range and maximize your damage (videos will be below!).

Ability Tips:
This is Zoe's primary damage tool, so landing it is very important. Late game, this ability one-shots almost everyone, and severely dents everyone else, even tanks. The trick is actually landing it, which we will discuss in the Tutorial section.
Learning to thread the needle is very important to successfully use this ability, and coupling it with Flash and Portal Jump will take a lot of practice! Ability Insight:

• Limited Inventory: Although Zoe can pick up lots of Spells, some restrictions apply! A list of Spells she can pick up are below. Spells marked with red are spells that do not drop from minions, and can only be picked up when an enemy uses them. Spells marked in green drop from both minions and enemies.
  
  Spoiler: Click to view

  
  Flash
  Heal
  Exhaust
  Cleanse
  Ignite
  Teleport
  Ghost
  Clarity
  Barrier
  Smite / Chilling Smite / Challenging Smite
  Everfrost
  Hextech Rocketbelt
  Hextech Gunblade
  Youmuu's Ghostblade
  Blade of the Ruined King
  Locket of the Iron Solari / Circlet of the Iron Solari
  Redemption / Salvation
  Twin Shadows / Frost Queen's Claim
  Shurelya's Battlesong / Talisman of Ascension
  Zhonya's Hourglass / Stopwatch
  Wooglet's Witchcap / Wooglet's Witchcrown
  Turbo Chemtank
  Randuin's Omen
  Predator
  Mark/Dash
  Although you cannot pick it up in any way, Hextech Flashtraption will trigger the movement speed and bonus damage from Spell Thief if you fully channel Hexflash.
  
• Really Limited Inventory: There are some things Zoe can't pick up at all! A list is below.
  
  Spoiler: Click to view

  
  Edge of Night
  Knight's Vow
  Seraph's Embrace
  Mikael's Blessing
  Zeke's Convergence
  Ohmwrecker
  Bilgewater Cutlass
  Zz'Rot Portal
  Banner of Command
  Quicksilver Sash / Mercurial Scimitar
  Gargoyle Stoneplate
  
  
• Magic Missile Massacre: Upon using Spell Thief, Zoe gains a burst of movement speed and gaisn three Magic Missiles that home in on her nearest target. If she is currently empowered by More Sparkles!, these bubbly missiles become instant-contact lasers instead! Zoe gains these effects both when using Spell Thief or when using her own Summoner Spells. Zoe does not gain these effect when using an item from her inventory.
• Daisy Chain: If Zoe has two of the same Spell, pressing either one will always use the Spell Thief copy, preserving the original copy. For example, if you have Flash as a Summoner Spell and as a Spell Shard, no matter which one you press, you will always use the Spell Thief copy.

Ability Tips:
This is possibly the most versatile ability in League of Legends. By taking advantage of your opponent's spells, you can use their powers against them. Learning when to dive or Portal Jump in for a Spell Shard, when to chain multiple Flashes together, or when to pop that ill-gotten Redemption is part of mastering Zoe. This is her hardest ability to really learn and control. You must be adaptable to any situation and react instantly as the battle unfolds, picking up Spell Shards and weaving them together with your other abilities. The uses for Spell Thief are literally endless, and I will discuss the many possibilities in the Tutorial section. Ability Insight:

• A Dream To Some: This ability first slows the enemy before putting them to Sleep. Although they are unable to act while Asleep, they will awake upon taking any damage.
• A Nightmare To Others: While Asleep, the next ability or attack deals bonus True Damage equal to or below the initial amount dealt by Sleepy Trouble Bubble on impact.
• Terrific Soporific: Although the range of Sleepy Trouble Bubble is rather short, it can be extended by throwing it through walls or terrain, ensuring it passes over the entire length of the obstacle.
• Night Light: Enemies that are put to Sleep are visible through the fog of war, revealed by telltale butterflies and an accompanying jingle.

Ability Tips:
The ability that started it all. This ability is what allows Zoe to burst so hard and siege so well. By utilizing choke points and predicting your opponent's behavior, you can land a Bubble and snip them from a mile away using Portal Jump and Paddle Star.
A downside to this ability is that there is a delay to the Sleep effect, meaning high mobility opponents can weasel their way out of range of your Paddle Star.
Another downside is that any damage will wake up your sleeping opponent, including attacks from your allies or monsters. Your entire combo can be ruined by a rude ADC! Ability Insight:

• Through the Horizon: If Zoe jumps into a location that she has no vision in, she'll gain a moderate amount of vision around her and over walls, similar to Quinn's Heightened Senses.
• Bend the Rules: If Zoe accrues any crowd control or targeted effects while using Portal Jump, she will always complete the return back to her original position, no matter what.'
• Portal Collapse: Zoe is not invulnerable during her teleportation, and any damage-over-time or area-of-effect damage will still hit her and affect her (like Jinx's Super Mega Death Rocket! or Ignite).

Ability Tips:
A pseudo-mobility skill, this is Zoe's primary engage tool that she uses to achieve maximum Paddle Star damage, land a long-range Sleepy Trouble Bubble, or finish off an enemy with More Sparkles!.
Although it seems like a fail-proof ability, your return location is very heavily telegraphed. Savvy enemies will snare and kill you as your return, making timing and planning very important for this ability to function properly.

Zoe likes AP, and doesn't really benefit from much else. Prioritize high-damage items like Rabadon's Deathcap and Morellonomicon, but remember to leave room for defensive items if you get slammed!
These three items comprise Zoe's core loadout. We take Luden's Tempest for the huge increase in mana and the bonus CDR, we take Lich Bane because it almost doubles her passive damage and provides bonus MS and even more mana, and finally we take Morellonomicon because it gives us flat Magic Penetration and health, which we will need if we want to survive during team fights.

For boots, you can take any boots you'd like! I prefer Sorcerer's Shoes or Mobility Boots, but if you're against a high-CC comp or a high-AD comp, opt for those Mercury's Treads or Ninja Tabi instead.
Zoe's item pool favors high-damage items, but has room for defensive options if you need them. Typically, you'll go right into damage items if you're doing well, but don't be afraid to play behind and grab a Zhonya's Hourglass to prevent the Zed from Deathmarking you.

Other options like Everfrost or Hextech Rocketbelt work as well if you need to add some movement or CC to your arsenal.

My final build for Zoe is usually this: