This is mostly for just personal reference, but this is how I play Janna so I thought I would share. This is quite the fun build, nice farm, harass, support and you can really dish out the damage.

The basis for this guide came from me using 0wned Heh's guide found here:
AP Nuke Janna Mid? What..?

I switched from a more support heavy Janna to an AP styled one and have been having more success and fun by starting out with this guide.

Build Overview

The first build is AP-centered for good poke damage and utility. This build is not recommended for use in ranked games and is intended to be more of a different way to play Janna.

Build two is a more conventional support build based around keeping your carry alive and aiding your team.



The Orianna and Lux builds are a work in progress!

BUILD ONE: AGGRESSIVE UTILITY

For items get Meki Pendant first and a couple of mana pots. Without mana regen runes, you will sorely need these until you can build up your items. If you tend to be more aggressive or find you are taking a lot of damage early on, swap one of the mana ones for a health one instead. I like building up to Archangel's Staff first for the early mana regen and to build up its passive.

Either get Boots before the staff, or you can wait until after to get them if you feel you don't need them because of Zephyr. I go for Mercury's Treads for the tenacity, but Sorcerer's Shoes are a good fit if there isn't reason for the stun reduction.

The next item is by preference. If the game is going really well, go for Rabadon's Deathcap. If you feel you need more survivability and utility, Rylai's Crystal Scepter is a good choice since it provides great AP, health and a slow to basic attacks, great for early game survivability and for when you throw a Howling Gale into the enemy team during fights. Lead these into a Void Staff when the timing is right.
As well you can go defensive if the game just isn't going for you. I like taking a Frozen Heart as the mana goes well with Archangel's and the cooldown reduction and AoE slow are great utilities. However, if the enemy team is not so AD heavy, you should use a Banshee's Veil as a good alternative.

For alternatives to defensive items you don't need or if you choose to not take one of the previous items, here is a list of possible choices:
Will of the Ancients: Great support item for an AP heavy team
Rod of Ages: A great all-around item, make sure if you are going for this get it early!
Lich Bane: The reason I do not get Lich Bane is because it is outclassed by other items in the stats it provides. The magic resist and mana are not comparable to Banshee's which also provides health and the movement is not really needed. As well, you will not be using abilities very often in range to also hit with many basic attacks so the passive is wasted. I feel that it does not go very well with a champion like Janna, though some guides do recommend it. You can always choose this instead of Rylai's since it provides basically the same utility in the form of movement speed instead of a slow to your attacks (though you would lose the AoE quality to your gales) and mana instead of health for an Archangel buff if you don't feel too squishy. I would recommend this if you choose to pass up on a defensive item in a game that is going well for you and you happen to have the cash.

BUILD TWO: HEAVY SUPPORT

Here I start off with a Faerie Charm to save room for potions and wards. This allows many setups: 2 health pots and 3 wards; 1 health pot, 1 mana pot and 2 sight wards (my preferred); 2 health pots, 2 mana pots and 1 ward.

Rush the Philosopher's Stone for the gold/5 to maximize its worth. Next I build into a Tear of the Goddess. I can't do without this item, it's mana use is too godly and it allows sticking around for long periods of time in lane and still have mana for moving around to assist teammates.

Next is your aura item. While I have Aegis of the Legion here, it can be replaced with a Zeke's Harbinger or Will of the Ancients depending on team composition and how well you are doing.

For boots I like to have Ionion Boots of Lucidity but having Mercury's Treads may prove more useful against teams with a lot of CC or AP.

I get a Shurelya's Battlesong next since it's active is very useful for initiating, chasing and escaping during/after team fights. I also turn my Tear into a Archangel's Staff.

For my final item taking a Banshee's Veil is a safe bet if you find you are dying or being focused a lot. If not, going for a Rabadon's Deathcap is a better bet to increase you damage, shield and heal. Janna has awesome AP ratios, remember!

Also remember to always keep a spot in your inventory open for wards! Map awareness wins games!

For runes I go flat ability power seals and glyphs and flat magic penetration for marks and quints to boost my harass and farm early.

Another set-up I like is armor seals and magic resist glyphs for an aggressive lane phase or if you just find you are too squishy early game.

As a pure support, build armor seals to protect you from the enemy carry, mana regeneration glyphs because you are oh-so mana hungry and health marks and quints for that extra damage absorption. Swap the quints for gold/5 by preference.

Flash

Flash provides a great way to escape messy situations, or get into ideal positioning fast. It can also prove to be an interesting way to place yourself for a good Howling Gale or Monsoon!

Clairvoyance

Clairvoyance is the perfect utility spell. It has saved me and my teammates from ganks and secured our own ganks time and again.

Heal

Since Janna lacks her own reliable heal because of Monsoon's large cooldown and requirement of level 6, Heal makes a good choice. If someone else has Clairvoyance (Build One) or if you feel like you'd prefer it over Flash (Build Two), take Heal.

Ignite

Ignite makes a great way to lock down healers and secure kills and can be taking along with Flash in build one.


SOME ALTERNATIVES:
Clarity: Janna is very mana hungry and dependent, but that's what all the regeneration runes and items are for. Later on, this will prove to lose its usefulness. Take it and lose some the Archangel's in build one if you feel your money can be better spent elsewhere.
Garrison: Janna's Eye of the Storm ability can target turrets. Pop one on, pop garrison and watch the tower destroy your enemies. If you are quick and clever, you can even Flash behind a nearby champion when the time is right and Monsoon them into it for added devastation.

POOR ALTERNATIVES:
Ghost: Zephyr passively allows Janna to ignore unit collision and Ghost can't be used in as many situations as Flash (ie: Flashing over walls or into strategic positioning).
Cleanse: Janna is very susceptible to CC, however since she has plenty of escape tools, I don't feel this is a good choice.