Ok, Im not a super long time Janna user, but Im a fan and have been using her in mid-ish tier play for a while, and successfully.
The first thing you have to realize is that you dont pick Janna to get the kills, nor as common belief do you pick her to save lives. She is a support, and a good Janna can save lives, but that isnt her main purpose. She works best as a pusher, securing an early game win, or providing enough harassment to allow your carries to enter end game without much enemy advancement.
Lets go through each of her abilities:
Q--This is her best ability in my opinion. It is a long range, low cd, harass, and stun. If you build AP (not too smart) it does meh damage. But it is best used t either stop groups pushing in their tracks for a few seconds, or to siege turtles from afar. A Janna pushing should almost always have this spell on CD as she wants enemies further down the lane to always have a tornado coming towards them (though she wants to save the CD if it feels like a gank set-up and she might need that tide turning mass stun). Overall, this spell will break pushes, and make pushes.
W--This is her second best ability. Again, never really go AP on Janna, but this spell does pack a punch early. More importantly it allows you to somewhat kite if they come close to you, or it allows your team to secure a kill due to the snare.
Its even bigger importance is the speed boost it grants/permanent ghost collision. Janna can almost always land this spell because it gives her pretty much more speed than most heroes, and creeps are a thing of the past for her. It also makes it so only the best chasers can really kill her.
Use the active sparingly though as it really is not to just pump out dps/snare, but only when the snare is needed.
E--Umm, well, it is a good spell, but I almost never get it until the end (though unfortunate circumstances might change this). Lets talk about when you dont get it until the end. This spell gives a very decent shield early game, and a damage boost that is significant as well that early. However, the power of the other spells foreshadows this. Also, the shield is more of a lifesaver, not a constant provision (early game you wont be able to spam 2 spells for long, and Q is much more important). What I end up using the shield for end game is just to provide a bit of a buffer for squishers, or if I see someone dominating, slap it on for an extra 40ish damage on their attacks (enjoy it Yi).
The few times you do get it early. You get it early in 2 situations. The first is if your lane-mate is a moron. Janna will never solo, it is a waste of a solo spot, and she will always be partnered for max effect. If you happen to get a braindead, or just super cocky moron in your lane, you might have to level shield early. You wont need W because its likely the other team is in more of a position to make kills due to your lanemate playing overly risky. And my rule of thumb is this, if youre playing Janna, you automatically have 1 less dps on your team, you cant afford the other players to suck; or conversely, you cant have your enemies own. Your shield can mean the difference between your partner feeding and not feeding. This in turn first makes it so he gains more gold/exp which will help later, and means that the enemy gets less gold/exp, win/win except that you dont have a more useful spell leveled up.
The second time you choose shield over Zephyr is if you are getting outpushed. This does not happen often, but if you lane against those special combos, think Soraka + Alistar, you will be pushed toward your tower. Again, you wont be in the position to be getting kills, but you will be getting pushed in. Janna's shield is one of the better IMO since it can be used on towers. Even a strong pushing team doesnt plow down that tower with just the usual lane pair, and they do so bit by bit until mid game. This means with the extra shield on the tower, you can have their bit by bit either do nothing, or very very little. This is critical as you do not want to lose towers if you can help it. Countering a pushing team early means you won late. So if it means saving towers, well, you have to level up shield.
R--I think this spell is overrated. Does it have the potential to save lives? Yes. Does it often save lives? No, not really.
The primary way this spell saves lives is by properly using the knockback. You can put a great deal of distance, and hopefully an initial tick on your ally, between the enemy horde and your dieing ally. This will in most cases save them, unless the ally was snared/stunned in which case the distance is not great enough.
Early game the heal will allow allies to fight within the AoE and be temporary kings. The heal is powerful eary (not that it is weak late), it is enough to really give AoE invul--close to. Late game, it will definitely help a lot, but dont count on it as much.
The real reason this spell is very nice is because it is a short CD AoE heal. During pushes, you have a clash and didnt need to burn it there? Well, get to the next building and pop it. You then have a full-ish HP team. This spell is much moe of a pushing aid, than a save your life aid.
In most Janna games you will hopefully win early, because she really is a pushing beast, as such, my item list for late game isnt as developed.
Start with the hp/mana regen pieces of "Haunting Guise." When you have 1200 gold return and finish Haunting Guise along with boots 1.
What does Haunting Guise give Janna, a non-dps caster? Well most importantly, cheap early game regens. She needs something to fuel her Q spam, Guise fills this nicely. The HP regen is also not too bad if you constantly run through creeps to land a hit and then back (she cant leave the lane and needs the hp regen). The CD reduction is really what seals the deal after the first 2 stats, more Q means more enemy QQ. The MPenn is just icing really, since you dont go AP on Janna, it pumps up her spells damage, as well as allowing it to still hurt creep waves when they start to get tougher.
Next up is a t2 boots. This really changes depending on the team, but you have a choice between Merc and Ninja. Always lean toward the Merc just because of the 40% CC reduction. But if the enemy has a Yi, Ashe, or some other heavy hitting physical line-up then grab Ninja as they do actually provide a good amount of survival.
Your final core item comes up only if you have been successful in farming and pushig early, Innervating Locket. This item is only useful if you are pushing with your team, if they havent learned to work together this item wont mean much (but then again youll likely lose anyway).
What this item does is 2 fold. First, it gives you much much needed HP along with much useful Mana. Survival is important on Janna because she needs to make her presence known with Q spam and team utility.
Second, its passive allows you to more easily push deeper turrets. Lets face it, the second turret is harder to push due to more ganking ground, and the third is even harder due to the mass turtle effect. Locket allows your team to stay in the action longer because they are constantly getting heals and mana regens. it doesnt seem like a lot, but until you get to late game, it adds up to a whole lot. You can try a game without this item, then one with, you will notice the difference. That said though, this is not a late game item except for a gorified mana regen with a non-passive cataylst.
After these three items you really should be finishing the game. Janna provides an incredible team pushing element that is very hard to counter if your team works i unison. But if it doesnt, the key to Janna late game is survival+ team support.
Items like Frozen heart are great, they not only help you live (if a physical team), but they benefit your team too due to the debuff. Likewise, Abyssal is also nice if versing a heavy magic team. Dont be selfish with items, you are nothing without your team.
If they notice how much of a threat you are, picking up more defense items above along with RoAges is nice. The RoAges is one of my favorite items. It offers plenty of HP, and even more mana (you probably dont even need much anymore). And the AP boost is always nice to just tack on (more powerful spells).
I know people mix and match, I always pick up the same 2; I get TP and Clairvoyance. TP is absolutely a must. You dont have time to waste running around. And especially as a pusher who depends on making the best of a small amount of time, you need to go from lane 3 to lane 1 asap. Also, it is just nice to be able to BP, refill hp/mana, and TP back to a hard push.
Now lots of people dont like Clairvoyance anymore. TBH, it is still a very useful spell, its just not useful enough to waste a spot if you are say Annie or Ashe. Luckily, Janna is all about helping the team. Clair is amazing for 3 things. First, you will never get early game ganked from the brush, simply clair the brush for both side lanes (yes the CD is that short), or you can clair your brush, and thendragon area if you notice a proper lineup (btw, Tornado cutting across a lvl 1 team doing dragon is really powerful). Second, you never lose sight of fleeing enemies. Third, you are way way more prepared for ganks; feel a gank, check the likely spot and give yourself a whole lot of time to fight them directly or run the hell away.
Some people might say Rally is better for a push team, but 3 things. First, other heroes can get rally as it typically is a nice spell for phys dps to pick up. Nobody else except you will pick up Clairvoyance. Second, sure it is nice, but Rally can be used only a few times, and with Janna, if youre going to get a tower down, whether you had Rally or not isnt much of a diff. Third, being harder to gank is important since you are early game dependant. 5 man ganksquads depend on getting kills, deny them that and they just wasted a lot of lane time, while your team didnt. Just remember that map control is important, and Clairvoyance does help.
Spam Q toward the clash of creeps. Make sure to hit all the creeps. You might be tempted to aim it to harass enemies, but dont. Targetting enemies is wrong for a few reasons. First, this spell is relatively easy to dodge. The onyl times yu should expect this spell to land is when the enemy is playing risky, or dumb luck; a focusing player will likely never be hit unless they put themselves ina risk situation. Second, not targetting creeps means that you first dont get gold, and second, you are not pushing the lane.
By targettting their creeps you also target yours. This means that if they want to farm, they have to always dodge your tornados. Not overly difficult with ranged, but melee do have a harder time.
Dont forget with Zephyr learned you can easily also run through creeps and bop them on the head if they still stick around tryign to hit creeps. Also, Zephyr hits hard early, take advanatge of it if you can get a kill by using it.
If you down your first lane tower, switch to the solo lane and do the same there.
You should definitely have downed your tower here. This is when you constantly move around the map until all the first towers are down. Hopefully the enemy cant say the same. This means you have a snowball advantage, use it.
You should have your locket by now. Depending on how strong your enemy is you might first need to set up a gank, but if you feel they are not strong enough to pose a threat in 5v5, just gather your team in one lane.
Now you push down. Your goal Mid game is to end the game. You are currently at your pushing peak. Your team is hopefully at a very nice exp/gold advantage. Focus on one lane until its base tower is down.
If you can secure an inhibitor, you will likely win. The earlier you down an inhibitor, the better. Super creeps are very strong, especially if the enemy is caste heavy (borderline gg there). One inhibotor may mean victory.
Once you get the first inhib down you have two options, nexus or another inhib. This choice depends on your enemy. If your enemy can barely handle the super creeps coming at them from the one inhib, meaning it is still relatively early, push a second inhib. 2 lanes of supers is practically mid game death. If its later on, almost late game, straight to nexus, that second inhib attempt might just give your enemy enough time to rally up and stop you, meaning that your push died and you didnt inflict enough damage to secure a win.
This is just like mid, except you have to be more cautious about your spells. A Yi running down all crazy eyed means you need your tornado to stun. But a Yi sitting next toa base tower means siege cannon as normal.
Late also means that you should straight for nexus unless you can super easily BD from the lane you ppenetrated.
You are pretty weak late game, so make sure you play especially well.