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.
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.