The way you teamfight varies with many more factors than the way you lane does, and there is little correlation between the way a support lanes and the way they teamfight---even for the same support but in different teamcomps. This makes it somewhat harder to describe how to teamfight as a support, whereas for an ADC it's relatively easy to say "attack the most valuable champion that you are able to target without putting yourself at too much risk relative to the reward from getting that particular autoattack in." However, for nearly every support and nearly every teamcomp, either your first or second goal in a teamfight is to get your carry to kill their bruisers without dying himself. The way you accomplish this goal is what varies.
Before I continue, I want to make an important point about CC that's in fact so important that I'll probably give it its own post sometime soon. CROWD CONTROL CAN ACCOMPLISH TWO THINGS. IT CAN STOP YOUR OPPONENTS FROM DEALING DAMAGE, AND IT CAN ALLOW YOUR TEAMMATES TO DEAL DAMAGE. IT IS THEREFORE MOST EFFECTIVE WHEN IT'S DOING BOTH AT THE SAME TIME. It's not always necessary to use your CC to do both at the same time: for example, it's 100% worth it to stun their Shen before he can get his taunt off on you so that you can flash over the wall to safety when you're running away through their jungle. Similarly, if you catch their support out while he's warding dragon, then you should use your CC to keep him there while your team quickly closes in for the kill. But in some sense these are the exceptions that prove the rule, because in both cases you're trying to swing a death in the favor of your team and it's not an option to have damage being dealt in both directions. That beautiful Sona flash-ult on 4 of the enemy team means nothing if you have no follow-up damage, and the Taric stun on the exhausted Jax whose jump is down anyway could have been used on the Riven who's currently autoattacking your carry, even if Riven is somewhat weaker than Jax.
I'VE NEVER PLAYED THIS HERO BEFORE WHAT DO
Here's a quick breakdown of what your default goal should be unless you have a specific reason to do something different for most common supports. For each champion I list two common sets of things you might want to do; the trend to notice is that one of them is almost always "keep your ADC alive," which is your goal when the rest of your peelers can't handle all of the enemy's divers. This might happen because you don't have any other peelers or it might happen because the enemy has two strong divers and your carry's other peeler can't handle both of them without a significant amount of help from you. In soloqueue, you're usually going to want to go with the "Keep your ADC alive" path because people don't tend to peel enough.
What does it mean to peel for your carry? A brief definition is that
peeling for your carry means stopping the enemy bruisers/divers from
being able to kill your carry, not by doing damage yourself, but by
letting your carry survive for long enough without being disabled to
deal damage. Basically every form of CC can be used as peel: Blitzcrank
or Nautilus can grab an enemy away from his carry; Nunu can slow
someone like Udyr who just wants to run at his carry; Soraka can silence
a Singed before he can fling her carry; snares and stuns can completely
stop gapclosing briefly; and blinds can stop any autoattack damage from
being dealt. Knockups and knockbacks are especially important because
they interrupt dashes like a Leona E or a Jarvan E-Q combo. This makes
Janna one of the best champions in the game at peeling and a very good
pick against hard-initiate comps.
Alistar: (1) Peel bruisers off your ADC by pulverizing and/or headbutting away. Use your ult to break CC or if you get focused. (2) Use your W-Q combo on the (majority of the) enemy team while ulting to mitigate damage taken.
Blitzcrank: (1) Pull someone squishy and use all of your CC on them to get the kill before they can deal any damage. (2) Use all of your CC to peel for your ADC.
Janna: (1) Use everything to peel for your ADC. (2) Flash-ult their out-of-position squishy into the middle of your team and use your other skills to stay alive. But actually, just use everything to peel for your ADC.
Leona: (1) Use your ult on enemy squishies and your other skills to peel for your ADC or maybe even use all your skills to peel for your ADC. (2) Dive the enemy ADC.
Lulu: (1) Ult a diver and use your other skills to peel for your ADC. If anyone on the enemy team has stealth, drop an early E on them!!!! Even if someone has an oracle!!!! (2) Use everything to peel for your ADC. Note that putting a W on your ADC is akin to peel because it makes your carry able to kite better.
Lux: (1) Shield your entire team, use your Q and E to help your ADC kite, and ult for damage across their carries. (2) Shield as many people as possible, and use your Q and E to CC the enemy squishies. It's unlikely that you want to use your E for damage; usually the slowing field is better, but if you somehow got fed early then you can just act like an AP carry.
Nami: (1) Use your ult to knock up as much of the enemy team as possible and your Q to CC divers. Use your E and W to help your carry kite and survive. Try to get your carry to have the extra 30 MS for as long as possible (i.e. use your E and then wait 1.5 seconds to W, unless he needs to be healed NOW). (2) Use all of your CC on the entire enemy team.
Nunu: (1) Make sure your ADC is going to be blood-boiled for the next 3 seconds and channel your ult in the middle of the enemy team when they are unable to CC you immediately to interrupt it. Preferably ice blast an autoattacker first. (2) Do the same, but channel your ult underneath your ADC as a form of peel. If you're too low to channel your ult safely like this then stay back and make sure you are able to continue blood-boiling your ADC.
Sona: (1) Ult the enemy team and get a W chord off on a damage dealer. Use an E chord to help your ADC kite and maintain an appropriate aura (more on this in my post "How to play Sona!"). (2) Use everything to peel for your carry.
Soraka: (1) Use your silence to stop important damage from an enemy mage (this could be interrupting a channel), heal your carry soon as you can heal for the full amount and ult as soon as you can heal for the full amount on a couple important people or when you need to heal someone particularly important. (2) Use everything to keep your ADC alive.
Taric: (1) Use your stun on someone who's diving your ADC and ult and shatter them. Heal your carry as soon as you can heal for the full amount. (2) Use your stun to catch someone out of position or perhaps even dive the enemy carries.
Zyra: (1) Hit an E on as many people as possible and then drop an ult below it. Make slowing plants unless you have a lot of damage or a Rylai's, in which case you can make Q plants. This path is pretty much the same as what you would do if you built AP. (2) Use your E to help peel for your ADC, but you probably still want to try to drop your ult on as much of the enemy team as possible, especially if you have an AoE comp; if you need to drop your ult to peel for your ADC then Zyra probably wasn't a great pick (but still drop your ult to peel!!).
SUMMONER SPELL USAGE.
For the most part, your summoners will be flash-exhaust. If you have heal, then heal your carry as soon as you can heal him for the full amount and he's going to continue taking damage OR when he's about to die and needs the heal immediately despite being ignited or having grievous wounds applied for some other reason. If you have clairvoyance, then use your CV somewhere that your team needs vision, although you're unlikely to have your CV up during a teamfight---you probably engaged on them as the result of a good CV or got engaged on by them as a result of not having your CV up.
So we'll assume you have exhaust. Usually, it's best to just exhaust the biggest threat RIGHT NOW to your carry, so that he can survive for longer. This means the Irelia or the Jax that just jumped on top of him. Try not to use CC at the same time as an exhaust: ideally, you will CC first and then exhaust as the CC is ending. If your CC wasn't available at the time that you exhausted, then wait to use your CC until the exhaust is over, unless either (a) you yourself can't survive that long or (b) their bruiser is so scary that he can kill your carry through an exhaust (this could be because your carry is almost dead or it could be because their bruiser is really fed).
Exceptions to this are when their bruisers are significantly weaker than your carry and/or other peelers. If your carry isn't having any problem dealing with the 0-7 Nocturne who just dove him, but their 5-0 Vayne is ripping through your frontline, then it's acceptable to flash towards Vayne to get in range for an exhaust and then exhaust her. If you're doing this, MAKE SURE that either (a) her cleanse is already down or (b) if she cleanses your exhaust then you have other CC that you can throw on her now that her cleanse is down, and that this CC is sufficient to get past the 65% tenacity that she has from having just used cleanse.
Finally, you need to be very aware of cooldowns on gap closers when using your exhaust. If Irelia's Q is up and she's close to within 650 range of your carry but not any closer, it might seem like you shouldn't exhaust her just when considering the positioning. After all, she can't do damage to your carry nor can your carry do damage to her. However, if you aren't close enough to your carry that you will be able to exhaust her after she jumps to him, you need to exhaust her now. On the other hand, if her Q is down, exhausting her now would be a waste, as you can walk towards your ADC at the same time as she does and thus be able to exhaust when the full duration is useful. You should know the cooldowns of all bruisers' gap closers at max rank and try to time them so that you can use your peeling skills appropriately.
Sometimes this is irrelevant because you get engaged on without warning, but hopefully your team has enough wards out that this doesn't happen, and you should always be aware of your positioning and actively try to optimize it anyway. There are three major things that you can be doing at the beginning of fights/when you expect a fight to happen.
YOU ARE AN INITIATOR. Sona wants to lead with her ult. Zyra can lead with a snare. Blitz leans with a hook (attempt). In these cases you will want to be near the front of your team so that you can Shurelyas-engage or perhaps flash-engage with the Shurelyas on your jungler happening secondarily so that there's even less notice for the enemy team. After the initiate, find your carry and stick to him (probably; again, there are many variables that affect what you do).
YOU ARE A PEELER. Stand very close to your carry. Not directly on top of, but close. Make sure that you're not blocking your carry from kiting away, and instead try to be stopping their bruisers from reaching your carry in the first place. This is extremely important. If your carry doesn't have a phantom dancer, make sure that you're not blocking his movement path.
YOUR MOST IMPORTANT JOB IS TO BREAK AOE CC. This can be silencing a Galio ult, or it can be using a Crucible on your carry after an Amumu ult. In this case, you actually want to stand far from your carry. This way, it's difficult to catch both you and your carry and another important member of your team in the CC. If this isn't an option for some reason, then buy merc treads so that you can give at least one other member of your team the "tenacity" that your skill provides.
Next time I'll talk about ways to learn effectively from spectating games and watching streams, which will mostly be non-specific to support, but I'll put it in this series anyway.