In team fights, my Janna is pretty awesome. Aside from still trying to figure out the best uses for Monsoon (it's really hard to dive in, flash and monsoon them into my team, so as neat as that sounds I haven't been able to execute) I generally get off good Qs, Ws and Es so that I help my team rack up the kills.

However, my laning Janna is awful. Even in my best Janna games I tend to die at least 5+ times, and the majority of these deaths are in the first 15 minutes as I get dived and zoned out of lane. I tend to play aggressive supports, so perhaps I'm playing Janna wrong but I feel like I need to push forward, auto attack and harass the carry and adc, but, I don't do any damage.

What should I be doing with Janna in lane? Should I just be tossing E on my adc and tossing out the occasional Q when needed and a W if they're low? Why am I struggling so much with Janna....

Janna's more passive-aggressive as a support. She doesn't have much burst to speak of, but she has a ton of skills that can enable aggressive play by her partner. You want to practise shielding your ally when the enemy carry or support starts attacking them, nullifying the damage - this is Janna's 'sustain'. You can use her tornado and slow to counter engagement attempts made by your opponents, or to trip someone up that's out of position.

Generally speaking, you don't want to bother with casual right click or spell harrass. Janna's easily killed if caught, and playing aggressive against people that can blast you in the face is a bad idea. However, if you can safely deliver autoattack harrass, there's no reason not to.

If you're dying a lot, try playing more passively. It's more fun to be aggressive and in the face of your opponents, but depending on the matchup and skill of both sides, that's just not always a good idea. If you're trying to play aggressive against a Soraka Graves lane you're just going to waste your time and eat lots of Buckshot harrass, for example. If it's a lane with a Taric in it though, your tornado resets his attempted initiate - by simply knocking the enemy AD carry in the air to prevent him doing anything as a followup.

I love janna, but if you are playing 0cs support, Janna's shield is awful to level up. The only time to level up E beyond 1, is when you and your adc are dominating the lane and you're able to throw them some extra AD...but it's still not great.
As support janna, So put 1 into Q, and then don't level it until last, or second to last. The knockup it provides is many times great enough for it's utility, to not need to spend on alot more mana by leveling it up.
I usually start with:
Q,W,E,W,W,R,E,W,E,W W......
IMO, W is really awesome. it's a wicked slow, is relatively light on the mana for what it gives(and she can be a mana hog), and does a good amount of damage if leveled up right away (which also allows you to put off boots2 for a while as well).

Help your carry win trades in lane. Shield to either permit aggression or to negate poke (fast reflexes help here!!).

I find it difficult to initiate trades with Q since it's easy to see coming (and this also drains mana quickly). Save mana for shields and ult and instead use Q and W to END trades to disengage and win trades. Don't bother AA harrassing beyond levels 1-2 unless they are too busy wailing on your carry to attack you back.

Also, oracles is amazing on Janna since you're fast and can kite away if you're caught.

Max shield first.

I find janna is generally a weak laner. A lot of aggressive supports can burst her down quickly, and sustain lanes can out sustain her. Her strength lies more in mid and lategame when she can turn teamfights. I used to main her but with the nerfs I play Sona instead.

It is usually a better idea to play defensively - shield your carries in a trade and generally use your tornado and slow for the disengage. Assist jungler ganks by pink wards and providing the needed CC in a gank.

The trick with Janna is that she can run out of mana very quickly. ESPECIALLY if you use your tornados. Generally speaking, you shouldn't use your tornadoes to harass (or at least very rarely). Your main goal in lane with Janna is to mitigate damage. You watch the enemy carefully, and when it looks like they're going in for a poke, shield your carry (if it looks like they're going in for a BURST, like Taric coming in for a stun with Graves right behind him, a tornado may be more appropriate).

Even if the enemy chooses not to attack your carry after you shield them (smart enemies wont bother attacking while they see your shield on them) shielding your carry also gives them a temporary AD boost along with the shield. And any GOOD ad carry knows to use the shield's duration to poke the enemy. If you don't see your carry making proper use of it, you should remind them that it's basically free poke.

Now, in the case that your carry is simply not poking (which in many cases is the safe thing to do, especially in a burst lane where going in for a poke can mean death), you can poke a bit yourself. There a few ways to do this. The safest is to just shield yourself and land an autoattack or two on the enemy carry. I've even seen some Janna's run armor penetration or attack damage runes/masteries to give them more harass in lane. It's much more reliable and cost-efficient than using your tornado, and your shield will absorb the counterattack. You can use your W too, thanks to low mana costs. But generally you want to level your shield first to mitigate damage, which means you'll actually get a stronger poke using your shield and autoattacks, thanks to the AD boost.

Summary:
Q - Reserve for interrupting enemy initiations, escaping, and ganks/kills; not harass. Long cooldown means you want to keep this up until it's absolutely necessary to use.
W - Use it when you want to slow one particular enemy (you can harass with this too thanks to low mana costs, but you're likely to take some damage in the process). Remember not to use it when escaping from multiple enemies so that you can maintain your passive speed boost.
E - Use on ADC to shield/poke, or on yourself to harass. Level this first to shield as much damage as possible in lane.
R - Use for escaping, pushing AOE spells away, clutch heals, blowing enemies into team/towers, etc.