Support item: You can go pretty much any support item. Shard of true ice is obviously superior for Zyra because of the AP. However, you can use whichever you feel comfortable with.
Boot rush?: Yes. The meta of this (addiction) video game requires it for two reasons. First, dodging ''skill''shots in botlane. Second, for roaming.
Mythic dilemma: I almost always go Luden's. The only time I'd even consider this bootleg Liandry's is if the enemy team has 3 tanky champions (tanks, juggernauts or w/e else).
Second item-Morello or Zhonyas?: So Silver players will say 'obviously Zhonya's'. No. You are a support. You are very gold-limited. The question of Morello or Zhonya's is a real one, unlike the mythic dilemma above. So, If enemy has 2 healers - Oblivion Orb; If you are carrying your team - Zhonya's. Most games will end before you get to your third item. As Zyra, you should really not position yourself in teamfights or skirmishes where you need a Zhonya purchase.
Other items are purely situational and/or based on your taste.

Pros:
-Your damage is almost hidden. Players will usually focus on an actually broken champion in your team rather than play around with your plants. This gives you the carry potential.
-You can steal CS and not get blamed for it.
-Your CC is actually stupid.
-Your ult is based.
-You can hold waves if your team falls behind.
-Miss every ability, still kill them.

Cons:
-NO MOBILITY. Your position must be on a level of an ADC or a Soraka if you want to survive.
-You MUST abandon your ADC and team for survival. You are more useful alive and with 0 damage than dead and with 2 dead on the enemy team.
-You gotta practice some geometry for your plant spawning. Not as much as Vel'koz, but you do.

LANE:

-Harass, harass, harass: I don't care if they have an engage, another poke, an all-in or a healing support. You are going to harass both the enemy ADC and the Support. You are going to be fearless. I don't care how many times you die in normals for this, you need to learn how to harass and live.
-Don't hold your ulti, don't hold your ignite: Showing them that you're as insane as the champion you're playing, even if it means flashing and igniting at lvl 2, is part of the mind game. Even if dragon is up and your jungler died 1v2 toplane (a likely scenario) and you have no plants, just use your ulti and ignite.
-You don't need to hit your Q or E unless you're getting ganked: If you get ganked and miss your E on their jungler, hope you enjoy a grey screen.
-Avoid roaming: So Diamond Dogs from other lanes will say that roaming is what makes support viable in this dirt-ADC meta. Not for you. You want levels. I don't care how much you want to feed your Katarina mid, you will be the one killing the Zed in mid-game. Focus on yourself.
-WARD WHEN THEIR JUNGLE IS TOP: DO NOT F

---

WARD IF YOU DON'T KNOW WHERE THEIR JUNGLE IS. You have NO mobility.

MID AND LATE GAME:

-Play 3 champions back from your team(mate): You get squashed, you are needed. Let anyone go first, stay back and just press that W-E-Q-R when your team or theirs engages, then run, then come back when your cooldowns are up. However, our old friends the assassins, think of this too.
-If enemy has assassin, ward your path: I want you to think where you will go for securing an objective and I want you to ward. Buy a blue trinket if needed.
-You need map awareness: Every new champion riot released and will release can dash 2 of your screens. This isn't Rakan. You need to know where you and everyone else is at.
-You can contest/stall objectives on your own: Your W can reach over the wall of Dragon/Baron pit. I recommend sweeping it before doing this as it if decrease the chance of getting killed from 100% to 70%. Basically, use your W-E, W-Q, or even ulti to stall the objective if you don't have a bounty.