Hey, Minion11 here, this will be my first ever MOBAFIRE guide on the recently released (for now) Zoe. I am a support main and I'd prefer jungle or mid as a secondary role. There has been some major joy in playing this champion mainly because of her bubbly character and fun ability set. Sorry there aren't any champion match-ups, it's because I do not believe I am informed enough to make any of those decisions, it's honestly up to personal preferance and skills/capabilities (also I'm lazy). Now then, the guide part.

Max R whenever you can like always, I may have messed up where you get it (I hope I didn't though), but again like most champs prioritize leveling this.
The first basic ability you max should be E, because you are the support, protecting your ADC is one of your main priorities, it will also buff their damage as well. Leveling E also increases the double damage cap which is more helpful early than late game.
Secondly, max Q for the damage, you are a mage support and will usually be trying to do damage as well.
Lastly, the only ability left is your W so take that.

Keystone: Arcane Comet is a good choice because Zoe won't make much use of Phase Rush or the allied part of Aery.
Sorcery Path: Mainly CDR based so you can use her abilities more often, water walking is also helpful for when you want to get some wards down or have a small fight near the dragon or baron pits.
Inspiration: The runes in the inspiration path are used mainly to save gold for larger items like Lich Bane, plus the extra 10 MS from any boots upgraded from Magical should help you get around faster and maybe dodge a skillshot or two.

Starting Items:
Spellthief's Edge is going to be the best support item for Zoe, you'll be trying to deal damage to the enemy with your high damaging Q. Relic Shield is going to be more helpful later on because of its tank stats and it has the benefit of letting you last hit certain minions, like ones with spells that you can steal with your W. Make sure you have your passive up to make last hitting easier. Ironic that Spellthief's Edge is a worse choice for Spell Thief.

First Back:
Simple, wards, trinket, leave.

Boot Options:
The Boots of Mobility are your cheap option if you want to get to your stronger items faster, this choice is not recommended as you can get some meaningful stats from your other choices. Sorcerer's Shoes are the recommended boots for Zoe because she is meant to be a mage and these give her magic penetration.Ionian Boots of Lucidity give CDR a highly recommended stat for Zoe because it lets her spam her abilities more often.

Core Items:
Just upgrade your Sightstone into either Eye items, the one according to your starting item of course. Redemption is there because it adds for extra protection and how strong of an item it is. ZEKE'S CONVERGENCE is going to be the strongest item for any support Zoe, it's ability when linked to an ally is activated whenever you use your ultimate ability, which is generally on a 10-4 second cooldown.

Extra Damage:
Lich Bane: Strong item that works well with your passive allowing you to get more auto attack damage in.
Ludens Echo: Your R teleporting forwards and backwards charges Luden's Echo, and yes I tested this.
Void Staff: Simple yet again, for the extra magic pen if you skipped the Sorcerer's Shoes.
Rylai's Crystal Scepter: This will give you a slow which is helpful for chasing down enemies or running away from them.

Extra Defense:
Locket of the Iron Solari: This item is mainly for its active which can help your team against high burst champions.
Zhonya's Hourglass/Paradox: Freezing yourself is pretty cool and useful, keep in mind using this mid-ult will still send you back in your golden form, works with stopwatch too.
Banshee's Veil: Useful if the enemy team has a hard engage spell such as a Malphite ultimate.

I've been playing this champion nonstop since her release, and that's totally not an overstatement. I'll place the these in separate categories based on ability.

Q, Paddle Star:
This ability gains more damage the farther it travels in a straight line, try throwing it backwards and then moving to the enemy so you can get some nice bonus damage.

W, Spell Thief:
This ability will be used very rarely since you won't be last hitting minions, unless you got Relic Shield as your starting item, but when you do get the enemies to use their summoner spells, take them. In laning phase you should prioritize getting heal, then flash, then exhaust, then ignite, then barrier, then cleanse, or whatever would be more helpful in the current situation (God of the Forge, Ornn, forbid anyone to take smite for bot lane). Keep in mind, don't dive their turrent for a teleport, teleports are dropped where the channel starts.

E, Sleepy Trouble Bubble:
You can shorten it to STB; pros, half the syllables; cons, you don't get to say bubble. This ability is maxed first always as support Zoe and it's jelly packed with stuff. Try faking out your opponents by pretending to ward up your brushes, but instead sending an bubble through a wall to double Sleepy Trouble Bubble's range. Try saying that five times fast, double Sleepy Trouble Bubble, double Sleepy Trouble Bubble, double Sleepy Trouble Bubble, double Sleepy Trouble Bubble, double Sleepy Trouble Bubble. When the enemy ADC is doing that "walk sideways, auto, walk the other way, auto" thing, throw your bubble when they auto, but in the center of their patrol path, this way they'll walk into it out of habit or get caught in the trap portion. When you are running away from someone in the jungle, when you're in a narrow part, throw your bubble in the center to get the trap set and you'll have them walk around it if you didn't hit them in the first place. Remember that sleeping enemies take double damage from the next attack and wake up, so you can let them get their beauty rest or try killing them.

R, Portal Jump:
Use this ability to set up your Q's, by getting far away and casting R then Q, or to set up your E's, by hitting a far enemy with some CC. You can also portal jump to reach for you W's spell shards. In the ability's gray section, it says that Zoe can see over walls, use this to scout out a bit by ulting on the same side of the wall you are on. Ulting also allows you to reach that tri-brush on the red side from the river if you so chose to do so.

In your one second window of time you can...
a) Cast Q and E.
b) Use Q and auto.
c) Use E and auto.
d) Use an Item along that has a cast time lower than one second.
e) Use an Item and ability whose combined cast time is lower than one second.
f) Do any of the above while also using an instant cast summoner spell or item.

I don't know how to do these so bye I guess...