i. Madate
This is a throwback to the Ardent + Stoneborne Pact Zyra way back then. Rushing Imperial Mandate into Rylais is a strong powerspike. If you're versing a very melee comp where you basically have a guaranteed root, you can build Ardent/Staff right after Mandate for utility then build Rylai's after. Font of Life also gives you access to Putrifier instead of going Morello. This is my current go-to build, because you can build utility and still be useful when behind or pivot to a more damage oriented build that provides debuffs.

ii. Moonstone
This gives Zyra decent damage with added utility and survivability from the HP the items give. Ingenious Hunter makes her proc moonstone more, and it is highly adviced to build Demonic Embrace right after Moonstone for even more heals but it's okay to build situational! This build is best taken against heavy poke or when both teams have beefy comps so the fights will be long, and you can abuse Moonstone.

iii. Locket
This is the current meta build on Zyra. Rush Tier 2 boots, Steelcaps for very attack heavy comp, Merc Threads for heavy CC/magic damage poke comp. Then go for Anathema's rush if you can lockdown enemy ADC during laning. (e.g. Ashe + Zyra bot) and rush Locket if against heavy burst enemy. (e.g. Diana jungle + Katarina mid) I personally don't enjoy this build that much, but I take it when I'm against hard lanes or we have no frontline to sustain damage in fights.

You can basically mix & match damage, utility, and tank items with the 3 builds! Don't be afraid to try things out. If you snowball hard, you can still opt for the usual Liandry's build, but I really just like going for utility. :>

Enjoy!

I'll mainly talk about Mandate here because it's the most flexible. For Moonstone & Locket builds refer to the one above.

i. Starting Items
1. Spellthief's always! You are a lane bully.\
2. Pots ofc...

ii. Rush
1. Sorc shoes - honestly a powerspike... mpen + ms
2. Imperial Mandate - this is always my first item because it's cheap and a big early powerspike. It's like a supportive Luden's!
3. Rylai's - Though some games, I rush this when I really need to peel for my team. (e.g. Constant 4 man bot dive) Works really well with Mandate and Zyra's plants!

iii. Damage & Debuff
1. Demonic Embrace - I build this when I'm ahead and versing bruisers or tanks who stack HP and not MR.
2. Shadowflame - I build this into heavy shield comps, so think vs Karma, Lulu, Ivern, Yasuo, or Yone. Basically shields...
3. Void Staff - I build this vs tanks who build MR. That's it...
4. Chemtech Putrifier - I build this vs high sustain comps. When versing Yuumi or Soraka rush oblivion orb, sometimes I even build putrifier 2nd item when the ADC also builds sustain (e.g. Draven/Samira)
5. Mejai's - good when snowballing but ur squishy so high risk!
8. Cosmic - good for kiting and good with Rylai's!
7. Elixir - late game whatever

iv. Protect & Empower
1. Ardent - throwback to Stoneborn Pact hehe. Good 3rd item with on-hit ADCs (Ashe or Vayne)
2. Staff - Good 3rd item with spellcasters (Lucian or Ezreal) or when team is Full AP.
3. Mikael's - Situational. Build vs point click CC. or against Morgana/Leona.
4. Redemption - Situational. Build vs heavy poke team.
5. Knight's Vow - Situational. Build when ADC keeps on getting focused AND they are the source of damage.
6. Zeke's - Really good with Mandate. Provides so much damage buffs for your team.
7. Wardstone - when u need vision or when win condition is baron/dragon.

v. Sustain & Durability
1. Zhonya - Decent 3rd item good for Flash + R + Zhonya combo. Sometimes I build 2nd item when against Zed/Talon.
2. Frozen Heart - vs full AD team
3. Banshee's Veil - Decent 3rd item too. Good vs hook champs or zoe or morg or any catcher tbh...
4. Abyssal Mask - vs full AP team
5. Anathema's - sometimes i get this even when not running locket guardian when i need to lockdown a fed hypercarry.
6. Winter's Approach - very situational. i only build when i know games will be long and we have no frontline to sustain a bit of damage when engaging.
7. Elixir - lategame whatever