Pros

Insane damage output
Nice crowd control
Free wards
An extra AP carry in mid to late game


Cons

Will be focussed (this will kind of keep your carry safe though)
Slow move speed
You have to build some defensive stats on her
Doesn't offer as much utility as other supports

Armor seals are mandatory on Zyra, you'll really need them to survive.
greater mark of hybrid penetration Hybdrid penetration is great on Zyra, she has long range on her auto attacks and it will help her deal great damage in lane with her abilities.
Since nearly all supports deal some kind of magic damage and supports usually deal more damage than their carries in the early game, magic resist runes will help you out in the early laning phase. You can switch those with scaling runes if you prefer having more magic resist in the late game.
As a support, CDR is a very useful stat, because it makes (mis)using your spells more forgiving, and makes sure you can respond to any given situation when needed.
I really like the extra movespeed on Zyra, because her base movement is with 325 among the slowest moving champions. These will help you get in lane easier and will help you juke skillshots.
You take Zyra support for her zoning through great damage output, ability power runes amplify her raw power.

I will only discuss certain important masteries, since in my opinion some utility masteries are pretty general support masteries which every support should take.

Bandit is a really useful mastery, Zyra's kit has great synergy with the extra gold you'll get from landing some auto attacks. Also, if you got a thorn spitter out there, it will focus anything you auto attacked within it's range, which will help you with generating a lot of gold from your Spellthief's Edge as well!

Expose Weakness is pretty good, since your huge amount of aoe damage will ensure your team will proc the full effect.

Recovery is an optional mastery, it will only be noticeable in the early game. It will help you recover a bit faster from harass in the laning phase, which will give you usually the winning edge in trades.

Since you're supporting, you won't get as much gold as you would in the mid lane, so with Zyra support we have to buy low cost items unless she's doing really well. Also, since you're playing the support role, suppport your team as much as possible! It's VERY possible to get some kills with Zyra support, and while the extra gold for your team won't hurt, it isn't your primary goal.

You'll want to start with a Spellthief's Edge, Health Potion and a Mana Potion or Stealth Ward. Since the jungle in season 5 is way harder than it was in season 4 I don't feel like junglers got enough power to gank bot lane until you'll get your sightstone, so the extra stealth ward is nice, but you'll be able to deal more early game aggression with the extra Mana Potion. Important is that Zyra benefits so well from the early aggression, because she'll be able to zone the opponent and proc Spellthief's Edge effect, which will lead to an early sighstone soon enough.

About Sightstone, I cannot stress how important it is to get an early Sightstone! This item will give Zyra the much needed health while the free wards are so gold efficient!

For boots, Sorcerer's Shoes offer magic penetration, which is the most important stat to support Zyra. After laning you should have, or nearly have, a Haunting Guise. The magic penetration is what defines the damage output of support Zyra. Upgrading to Liandry's Torment late game will usually make Zyra the most damage dealing support there is. She'll probably deal the most damage of your whole team.

While Grasping Roots should be maxed first, it's not that good for poking, so put some points in deadly bloom before you start maxing Grasping Roots. Note that your plants scale with your level; their damage does not depend on maxing an ability. Putting a point in Rampant Growth should do, until level 14. Put a point in Rampant Growth at level 2, so you can start your ranged poke.

Flash is a must-have on Zyra. Use it to secure kills, use it to save your carry, use it to save yourself or use it for baiting purposes.
Exhaust is a very common support spell for supports, since the slow, damage reduction and armour shred will make you win a trade very easily. Use it to survive a gank, secure a gank/kill or use it when someone tries to all in on you or your carry.
Ignite is a viable, but less utility focused spell on Zyra, since you'll often poke a carry down to near-death. It's great for securing an early kill. The vision it provides will also help against stealthed enemies, but I'd rather use an Oracle's Lens or Vision Ward in these situations.

Zyra is a flexible support, since she's able to switch between a poke and kill lane. This depends entirely on the carry you're supporting. Some examples of poke lanes are:

Varus
Caitlyn
Jinx
Lucian
Kog'Maw

All-in lanes:

Vayne
Ashe
Graves
Quinn
Draven

When you're doing a poke-comp, focus on zoning the enemy. It's important to know when you need to create plants and when you shouldn't. Killing your plants will net your enemies a small amount of gold, which isn't ground-breaking, but try to not create plants when you don't need them. The difference between winning and losing a fight could be a potion.

An all-in comp is way more straight forward. Try to initiate a fight with Grasping Roots and a plant to slow your enemies. A bursty adc like Graves will be able to all-in the enemy you rooted, while you throw down another deadly bloom + Rampant Growth, to secure the kill.